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On the Kirchhoff matrix, a new Kirchhoff index and the Kirchhoff energy

Ayse Dilek Maden1, Ahmet Sinan Cevik1*, Ismail Naci Cangul2 and Kinkar C Das3

Author Affiliations

1 Department of Mathematics, Faculty of Science, Selçuk University, Campus, Konya, 42075, Turkey

2 Department of Mathematics, Faculty of Arts and Science, Uludag University, Gorukle Campus, Bursa, 16059, Turkey

3 Department of Mathematics, Sungkyunkwan University, Suwon, 440-746, Republic of Korea

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Journal of Inequalities and Applications 2013, 2013:337  doi:10.1186/1029-242X-2013-337

The electronic version of this article is the complete one and can be found online at: http://www.journalofinequalitiesandapplications.com/content/2013/1/337


Received:4 March 2013
Accepted:9 July 2013
Published:24 July 2013

© 2013 Maden et al.; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The main purpose of this paper is to define and investigate the Kirchhoff matrix, a new Kirchhoff index, the Kirchhoff energy and the Kirchhoff Estrada index of a graph. In addition, we establish upper and lower bounds for these new indexes and energy. In the final section, we point out a new possible application area for graphs by considering this new Kirchhoff matrix. Since graph theoretical studies (including graph parameters) consist of some fixed point techniques, they have been applied in the fields such as chemistry (in the meaning of atoms, molecules, energy etc.) and engineering (in the meaning of signal processing etc.), game theory, and physics.

MSC: 05C12, 05C50, 05C90.

Keywords:
Kirchhoff matrix; Kirchhoff Estrada index; Kirchhoff energy; lower and upper bounds

1 Introduction and preliminaries

It is well known that the resistance distance between two arbitrary vertices in an electrical network can be obtained in terms of the eigenvalues and eigenvectors of the combinatorial Laplacian matrix and the normalized Laplacian matrix associated with the network. By studying the Laplacian matrix in spectral graph theory, many properties over resistance distances have been actually proved [1,2]. Meanwhile the resistance distance is a novel distance function on graphs which was firstly proposed by Klein and Randic [3]. As depicted and studied in [4], the term ‘resistance distance’ was used for chemical and physical interpretation.

We note that throughout this paper all graphs are assumed to be simple, that is, without loops, multiple or directed edges. We also note that a graph G with n-vertices and m-edges is called <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M1">View MathML</a>-graph. Now, let us assume that G is connected and the vertices are labeled by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M2">View MathML</a>. By considering these vertices in the <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M1">View MathML</a>-graph G, in [5], the standard distance, denoted by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M4">View MathML</a>, between two vertices <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M5">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M6">View MathML</a> as the length of the shortest path that connects <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M5">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M6">View MathML</a> was defined. Moreover, again by considering <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M5">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M6">View MathML</a>, another distance (especially in molecular graphs), namely resistance distance, was investigated and denoted by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M11">View MathML</a> in such a graph G (see, for example, [1,6]). Let J denote a square matrix of order n such that all of its elements are unity. Then, for all connected graphs with two or more vertices, the matrix <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M12">View MathML</a> is non-singular with the inverse

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M13">View MathML</a>

After that, the resistance distance <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M11">View MathML</a> was defined in terms of X as <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M15">View MathML</a>[1]. In addition to this last distance, the matrix whose <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M16">View MathML</a>-entry is <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M11">View MathML</a> was called the resistance distance matrix<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M18">View MathML</a> which is symmetric and has a zero diagonal. As we mentioned previously, the concept of resistance distance has been studied a lot in chemical studies [2,6]. Furthermore, by considering the resistance distance of the graph G, the sum of resistance distance of all pairs of vertices as the equation

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M19">View MathML</a>

(1)

which was named the ‘Kirchhoff index’ [6,7], was introduced and investigated.

2 Kirchhoff matrix and Kirchhoff Laplacian matrix

In the following, by considering the resistance distance between any two vertices, we first define the Kirchhoff matrix<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M20">View MathML</a> as a weighted adjacency matrix.

Let G be an <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M1">View MathML</a>-graph. Then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M16">View MathML</a>-entry of the <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M23">View MathML</a> matrix <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M20">View MathML</a> is defined by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M25">View MathML</a>

(2)

We recall that the Laplacian matrix of the graph G is <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M26">View MathML</a>, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M27">View MathML</a> is the diagonal matrix of vertex degrees and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M28">View MathML</a> is the <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M29">View MathML</a>-adjacency matrix of the graph G. Using (2) for the definition of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M20">View MathML</a> and also using the Laplacian matrix, we can then define the Kirchhoff-Laplacian matrix<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M31">View MathML</a> of G as

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M32">View MathML</a>

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M33">View MathML</a>. For simplicity, let us label each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M34">View MathML</a> by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M35">View MathML</a>. Then it is clear that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M36">View MathML</a>

The eigenvalues of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M31">View MathML</a> are denoted by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M38">View MathML</a> such that the smallest eigenvalue is <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M39">View MathML</a> with eigenvector <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M40">View MathML</a>. Since we have assumed that G is connected, while the multiplicity of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M41">View MathML</a> is one, multiplicities of the remaining eigenvalues can be denoted by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M42">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M43">View MathML</a>.

After all, we can define a new Kirchhoff index in the form

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M44">View MathML</a>

(3)

where each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M45">View MathML</a> is as given in (2).

By considering <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M31">View MathML</a>, it is actually easy to rewrite the new Kirchhoff index, defined in (3), as a new form. In fact, this shows that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M47">View MathML</a> is completely determined by the Kirchhoff-Laplacian spectrum. In detail, by considering this new form of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M47">View MathML</a> (see (4) below), the ordering among the eigenvalues <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M49">View MathML</a> (<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M50">View MathML</a>) of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M31">View MathML</a> and the equality

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M52">View MathML</a>

we can obtain a new lower and upper bound (see (5)) for this new Kirchhoff index as in the following proposition.

Proposition 1For an<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M1">View MathML</a>graphG, let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M54">View MathML</a>be the Kirchhoff-Laplacian spectrum ofG, defined by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M55">View MathML</a>

Then a new Kirchhoff index ofGis defined by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M56">View MathML</a>

(4)

and bounded by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M57">View MathML</a>

(5)

3 On the Kirchhoff energy of a graph

It is known that there are quite wide applications based on eigenvalues of the adjacency matrix in chemistry [8,9]. In fact one of the chemically (and also mathematically) most interesting graph-spectrum, based on quantities in the graph energy, is defined as follows.

Let G be an <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M1">View MathML</a>-graph and let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M28">View MathML</a> be its adjacency matrix having eigenvalues <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M60">View MathML</a>. We note that by [10] these λ’s are said to be the eigenvalues of the graph G and to form its spectrum. Then the energy <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M61">View MathML</a> of G is defined as the sum of the absolute values of these eigenvalues as

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M62">View MathML</a>

We may refer to [11-13] for more details and new constructions on the graph energy. In view of evident success of the concept of graph energy, and because of the rapid decrease of open mathematical problems in its theory, energies based on the eigenvalues of other graph matrices have been introduced very widely. Among them the Laplacian energy <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M63">View MathML</a>, pertaining to the Laplacian matrix, can be thought of as the first [14]. We note that the theory of energy-like graph invariants was firstly introduced by Consonni and Todeschini in [15]. Later on, Nikiforov [16] extended the definition of energy to arbitrary matrices making thus possible to conceive the incidence energy [17] based on the incidence matrix, etc.

As in other energies mentioned in the above paragraph, we can define a new energy by considering the Kirchhoff matrix given in (2) as follows.

If G is an <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M1">View MathML</a>-graph, then the Kirchhoff energy of G, denoted by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M65">View MathML</a>, is equal to

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M66">View MathML</a>

(6)

where each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M67">View MathML</a> (with ordering <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M68">View MathML</a>) denotes the eigenvalues of the Kirchhoff matrix <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M20">View MathML</a>. Basically, these eigenvalues are said to be the K-eigenvalues of G.

3.1 Bounds for the Kirchhoff energy

In this subsection we mainly present upper and lower bounds over the Kirchhoff energy defined in (6).

The first result is the following.

Theorem 1LetGbe a graph with<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M70">View MathML</a>vertices. Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M71">View MathML</a>

whereκis the sum of the squares of entries of the Kirchhoff matrix<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M20">View MathML</a>.

Proof We have that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M73">View MathML</a>

By the Cauchy-Schwartz inequality, we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M74">View MathML</a>

Consider the function

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M75">View MathML</a>

Now our aim is to find the maximum value of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M76">View MathML</a>. To do that, we need to calculate the derivatives

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M77">View MathML</a>

By a simple calculation, it is clear that the equality <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M78">View MathML</a> implies <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M79">View MathML</a> and then, for this equality between x and y, it is also true that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M80">View MathML</a>

The calculations above actually conclude that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M76">View MathML</a> has a maximum value at <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M79">View MathML</a> and the required maximum value of this function is

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M83">View MathML</a>

Hence the result. □

The following lemma is needed for our other results that are given in this paper.

Lemma 1LetGbe a connected<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M1">View MathML</a>-graph and let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M85">View MathML</a>be theK-eigenvalues of G. Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M86">View MathML</a>

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M87">View MathML</a>

(7)

Proof We clearly have <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M88">View MathML</a>. Moreover, for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M89">View MathML</a>, the <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M90">View MathML</a>-entry of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M91">View MathML</a> is equal to <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M92">View MathML</a>. Hence, we obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M93">View MathML</a>

as required. □

Theorem 2IfGis a connected<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M1">View MathML</a>-graph, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M95">View MathML</a>

Proof In the Cauchy-Schwartz inequality <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M96">View MathML</a>, if we choose <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M97">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M98">View MathML</a>, then we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M99">View MathML</a>

from which

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M100">View MathML</a>

Therefore this gives the upper bound for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M65">View MathML</a>.

Now, for the lower bound of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M65">View MathML</a>, we can easily obtain the inequality

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M103">View MathML</a>

which gives directly the required lower bound.

We should note that there is the second way to prove the upper bound that can be presented as follows.

Let us consider the sum

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M104">View MathML</a>

By a direct calculation, we obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M105">View MathML</a>

It follows from (7) and the definition of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M65">View MathML</a> that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M107">View MathML</a>

Here, since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M108">View MathML</a>, we have <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M109">View MathML</a>.

Hence the result. □

In the following, we present a new lower bound which is better than the lower bound given in Theorem 2.

Theorem 3LetGbe a connected<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M1">View MathML</a>-graph and letbe the absolute value of the determinant of the Kirchhoff matrix<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M20">View MathML</a>. Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M112">View MathML</a>

Proof In the light of Theorem 2, if we show the validity of the lower bound, then this finishes the proof.

By the definition of Kirchhoff energy given in (6) and by the equality in (7), we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M113">View MathML</a>

(8)

Since, for nonnegative values, the arithmetic mean is not smaller than the geometric mean, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M114">View MathML</a>

(9)

After that, by combining Equations (8) and (9), we obtain the required lower bound. □

Theorem 4IfGis a connected<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M1">View MathML</a>-graph, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M116">View MathML</a>

(10)

Proof We apply the standard procedure (see, for instance, [18,19]) to obtain such upper bounds.

By applying the Cauchy-Schwartz inequality to the two <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M117">View MathML</a> vectors <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M118">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M119">View MathML</a>, where each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M67">View MathML</a> (<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M121">View MathML</a>) is a K-eigenvalue, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M122">View MathML</a>

Now, let us consider the function

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M123">View MathML</a>

In fact, by keeping in mind <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M124">View MathML</a>, we set <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M125">View MathML</a>. Using

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M126">View MathML</a>

we get that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M127">View MathML</a>

In other words, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M128">View MathML</a>. Meanwhile, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M129">View MathML</a> implies that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M130">View MathML</a>

Therefore f is a decreasing function in the interval

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M131">View MathML</a>

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M132">View MathML</a>

Hence,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M133">View MathML</a>

and so the inequality in (10) holds. □

4 Kirchhoff Estrada index of graphs

As a new direction for studying indexes and their bounds, we introduce Kirchhoff Estrada index and then investigate its bounds. Moreover, we obtain upper bounds for this new index involving the Kirchhoff energy of graphs. In order to do that, we divide this section into two cases.

We first recall that the Estrada index of a graph G is defined by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M134">View MathML</a>

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M135">View MathML</a> are the eigenvalues of the adjacency matrix <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M28">View MathML</a> of G (see [11,20-26]). Denoting by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M137">View MathML</a> the kth moment of the graph G, we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M138">View MathML</a>

and recalling the power-series expansion of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M139">View MathML</a>, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M140">View MathML</a>

By [23], it is well known that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M141">View MathML</a> is equal to the number of closed walks of length k of the graph G. In fact the Estrada index of graphs has an important role in chemistry and physics, and there exists a vast literature that studied the Estrada index. In addition to Estrada’s papers depicted above, we may also refer the reader to [27,28] for more detailed information such as lower and upper bounds for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M142">View MathML</a> in terms of the number of vertices and edges, and some inequalities between <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M142">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M61">View MathML</a>.

4.1 Bounds for the Kirchhoff Estrada index

For an <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M1">View MathML</a>-graph G, the definition of the Kirchhoff Estrada index<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M146">View MathML</a> can be given as

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M147">View MathML</a>

(11)

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M148">View MathML</a> are the K-eigenvalues of G.

Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M149">View MathML</a>. Then, similar to the <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M150">View MathML</a> case, the equality in (11) can be rewritten as

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M151">View MathML</a>

Thus the main result of the subsection is the following.

Theorem 5LetGbe a connected<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M1">View MathML</a>-graph. Then the Kirchhoff Estrada index is bounded as

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M153">View MathML</a>

(12)

Equality holds on both sides if and only if<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M154">View MathML</a>.

ProofLower bound. Directly from Equation (11), we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M155">View MathML</a>

By the arithmetic-geometric mean inequality (AGMI), we also get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M156">View MathML</a>

By means of power-series expansion and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M157">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M158">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M159">View MathML</a>, we clearly obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M160">View MathML</a>

Since we require a lower bound to be as good as possible, it looks reasonable to replace <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M161">View MathML</a> by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M162">View MathML</a>. Now, let us use a multiplier <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M163">View MathML</a> instead of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M164">View MathML</a>. We then arrive at

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M165">View MathML</a>

Now, for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M70">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M167">View MathML</a>, it is easy to see that the function

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M168">View MathML</a>

monotonically increases in the interval <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M169">View MathML</a>. As a result, the best lower bound for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M146">View MathML</a> is attained for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M171">View MathML</a>. This gives us the first part of the theorem.

Upper bound. By considering the definition and equality of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M146">View MathML</a>, we clearly have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M173">View MathML</a>

and then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M174">View MathML</a>

Hence, we get the right-hand side of the inequality given in (12).

In addition to the above progress, it is clear that the equality in (12) holds if and only if the graph G has all zero Kirchhoff eigenvalues. Since G is a connected graph, this only happens when <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M175">View MathML</a>.

Hence the result. □

In [29], by considering the maximum eigenvalue, Zhou et al. presented a lower bound for the reciprocal distance matrix in terms of the sum of the ith row of it. By the same idea, one can also give a lower bound for the maximum eigenvalue <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M176">View MathML</a> in terms of the sum of the ith row of the Kirchhoff matrix <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M20">View MathML</a> and for the number of vertices n. We should note that the proof of the following lemma can be done quite similarly as the proof of the related result in [29]. (At this point we recall that for simplicity, each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M34">View MathML</a> was labeled by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M35">View MathML</a> in Section 2.)

Lemma 2LetGbe a connected graph with<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M70">View MathML</a>vertices. Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M181">View MathML</a>

where<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M35">View MathML</a>is the sum of theith row of<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M20">View MathML</a>. Here, the equality holds if and only if<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M184">View MathML</a>.

Therefore the lower bound on the Kirchhoff Estrada index of the graph G (which was one of our focusing points) can be given as the following theorem.

Theorem 6LetGbe a connected<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M1">View MathML</a>-graph with<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M70">View MathML</a>. Then we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M187">View MathML</a>

(13)

In (13) the equality holds if and only if<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M188">View MathML</a>.

Proof As a special case of the theory, if we assume that G is a null graph <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M189">View MathML</a>, then for each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M190">View MathML</a>, we get <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M191">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M192">View MathML</a>. Thus <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M193">View MathML</a> and equality holds in Equation (13). In the reverse part, if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M194">View MathML</a>, then by AGMI, one can easily see that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M195">View MathML</a> and hence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M196">View MathML</a>.

As a general case, let us suppose that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M197">View MathML</a>. Therefore <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M198">View MathML</a>. We then have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M199">View MathML</a>

(14)

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M200">View MathML</a>

(15)

Now, by considering the function <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M201">View MathML</a> with its derivative <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M202">View MathML</a>, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M203">View MathML</a>, we easily conclude that f is an increasing function for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M203">View MathML</a>. Hence, from (14) and by Lemma 2, we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M205">View MathML</a>

(16)

This completes the proof of the inequality part of (13).

Now suppose that equality holds in (13). This implies that equalities also hold throughout (14)-(16). From the equality of (14) and by AGMI, we obtain <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M206">View MathML</a>. Since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M207">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M208">View MathML</a>, we must have <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M209">View MathML</a>. Thus G is a connected graph. Moreover, from the equality of (16), we have <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M210">View MathML</a>. Since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M211">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M212">View MathML</a>, by Lemma 2, G is a complete graph <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M213">View MathML</a>.

The converse part is clear, i.e., the equality holds in (13) for the complete graph <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M213">View MathML</a>.

Hence the result. □

4.2 An upper bound for the Kirchhoff Estrada index involving the Kirchhoff energy

Here, for a connected graph G, the main aim is to show that there exist two upper bounds for the Kirchhoff Estrada index <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M146">View MathML</a> with respect to the Kirchhoff energy <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M65">View MathML</a>.

Theorem 7LetGbe as above. Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M217">View MathML</a>

(17)

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M218">View MathML</a>

(18)

Equality holds in (17) or (18) if and only if<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M175">View MathML</a>.

Proof By considering the proof of Theorem 5, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M220">View MathML</a>

Moreover, by considering the Kirchhoff energy defined in (6), we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M221">View MathML</a>

which leads to (as in Theorem 5)

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M222">View MathML</a>

Hence we obtain the inequality in (17).

Another approximation to obtain an upper bound related to the relationship between <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M146">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M224">View MathML</a> can be presented as follows

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M225">View MathML</a>

which implies

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M226">View MathML</a>

as claimed in (18). By a similar idea as in the previous results, the equality holds in (17) and (18) if and only if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M175">View MathML</a>. □

5 Final remark

As it has been mentioned in some parts of the previous sections, it is well known that some special type of matrices, indexes and energies obtained from graphs play an important role in applications, especially, in computer science, optimization and chemistry. This section is devoted to pointing out a possible new application area in spectral graph theory by considering the Kirchhoff matrix defined in this paper. Although the problem mentioned in the following paragraphs would seem easy for some of the researchers, we cannot prove it at the moment and believe that it would be kept as a future project.

In [[30], p.537], the authors defined the Kirchhoff matrix over a loopless connected digraph, say D. In fact, by using the same notation as in this reference, we can define it as a matrix <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M228">View MathML</a> obtained from the incidence matrix M of D by deleting the row <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M229">View MathML</a>. After that, by an algebraic approximation over digraphs, it was depicted that K is a basis of the row space of M (such that each element in the basis set was called tension). According to Sections 7, 10 and 20 in [30], since there is a direct relationship between cycles and bonds in graphs and digraphs, and since tensions in a graph (or a digraph) are the linear combination of the tensions associated with their bonds, the authors produced the relationship between the Kirchhoff matrix over the digraph D and electrical networks (in Section 20).

In Equation (2) of this paper, we have defined the new Kirchhoff matrix in spectral graph theory and, as far as we know, there is no such study about it in the literature. Therefore, by considering the facts and results given in the previous paragraph, one can try to investigate a similar approximation to the relationship between this new Kirchhoff matrix over <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/337/mathml/M1">View MathML</a>-graph G and electrical networks.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors completed the paper together. All authors read and approved the final manuscript.

Acknowledgements

This work was presented in The International Conference on the Theory, Methods and Applications of Nonlinear Equations.

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