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 bounds1 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 nvertices and medges is called graph. Now, let us assume that G is connected and the vertices are labeled by . By considering these vertices in the graph G, in [5], the standard distance, denoted by , between two vertices and as the length of the shortest path that connects and was defined. Moreover, again by considering and , another distance (especially in molecular graphs), namely resistance distance, was investigated and denoted by 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 is nonsingular with the inverse
After that, the resistance distance was defined in terms of X as [1]. In addition to this last distance, the matrix whose entry is was called the resistance distance matrix 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
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 as a weighted adjacency matrix.
Let G be an graph. Then entry of the matrix is defined by
We recall that the Laplacian matrix of the graph G is , where is the diagonal matrix of vertex degrees and is the adjacency matrix of the graph G. Using (2) for the definition of and also using the Laplacian matrix, we can then define the KirchhoffLaplacian matrix of G as
where . For simplicity, let us label each by . Then it is clear that
The eigenvalues of are denoted by such that the smallest eigenvalue is with eigenvector . Since we have assumed that G is connected, while the multiplicity of is one, multiplicities of the remaining eigenvalues can be denoted by , .
After all, we can define a new Kirchhoff index in the form
where each is as given in (2).
By considering , it is actually easy to rewrite the new Kirchhoff index, defined in (3), as a new form. In fact, this shows that is completely determined by the KirchhoffLaplacian spectrum. In detail, by considering this new form of (see (4) below), the ordering among the eigenvalues () of and the equality
we can obtain a new lower and upper bound (see (5)) for this new Kirchhoff index as in the following proposition.
Proposition 1For angraphG, letbe the KirchhoffLaplacian spectrum ofG, defined by
Then a new Kirchhoff index ofGis defined by
and bounded by
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 graphspectrum, based on quantities in the graph energy, is defined as follows.
Let G be an graph and let be its adjacency matrix having eigenvalues . 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 of G is defined as the sum of the absolute values of these eigenvalues as
We may refer to [1113] 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 , pertaining to the Laplacian matrix, can be thought of as the first [14]. We note that the theory of energylike 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 graph, then the Kirchhoff energy of G, denoted by , is equal to
where each (with ordering ) denotes the eigenvalues of the Kirchhoff matrix . Basically, these eigenvalues are said to be the Keigenvalues 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 withvertices. Then
whereκis the sum of the squares of entries of the Kirchhoff matrix.
Proof We have that
By the CauchySchwartz inequality, we get
Consider the function
Now our aim is to find the maximum value of . To do that, we need to calculate the derivatives
By a simple calculation, it is clear that the equality implies and then, for this equality between x and y, it is also true that
The calculations above actually conclude that has a maximum value at and the required maximum value of this function is
Hence the result. □
The following lemma is needed for our other results that are given in this paper.
Lemma 1LetGbe a connectedgraph and letbe theKeigenvalues of G. Then
and
Proof We clearly have . Moreover, for , the entry of is equal to . Hence, we obtain
as required. □
Theorem 2IfGis a connectedgraph, then
Proof In the CauchySchwartz inequality , if we choose and , then we get
from which
Therefore this gives the upper bound for .
Now, for the lower bound of , we can easily obtain the inequality
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
By a direct calculation, we obtain
It follows from (7) and the definition of that
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 connectedgraph and let ∇ be the absolute value of the determinant of the Kirchhoff matrix. Then
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
Since, for nonnegative values, the arithmetic mean is not smaller than the geometric mean, we have
After that, by combining Equations (8) and (9), we obtain the required lower bound. □
Theorem 4IfGis a connectedgraph, then
Proof We apply the standard procedure (see, for instance, [18,19]) to obtain such upper bounds.
By applying the CauchySchwartz inequality to the two vectors and , where each () is a Keigenvalue, we have
Now, let us consider the function
In fact, by keeping in mind , we set . Using
we get that
In other words, . Meanwhile, implies that
Therefore f is a decreasing function in the interval
and
Hence,
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
where are the eigenvalues of the adjacency matrix of G (see [11,2026]). Denoting by the kth moment of the graph G, we get
and recalling the powerseries expansion of , we have
By [23], it is well known that 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 in terms of the number of vertices and edges, and some inequalities between and .
4.1 Bounds for the Kirchhoff Estrada index
For an graph G, the definition of the Kirchhoff Estrada index can be given as
where are the Keigenvalues of G.
Let . Then, similar to the case, the equality in (11) can be rewritten as
Thus the main result of the subsection is the following.
Theorem 5LetGbe a connectedgraph. Then the Kirchhoff Estrada index is bounded as
Equality holds on both sides if and only if.
ProofLower bound. Directly from Equation (11), we get
By the arithmeticgeometric mean inequality (AGMI), we also get
By means of powerseries expansion and , , , we clearly obtain
Since we require a lower bound to be as good as possible, it looks reasonable to replace by . Now, let us use a multiplier instead of . We then arrive at
Now, for and , it is easy to see that the function
monotonically increases in the interval . As a result, the best lower bound for is attained for . This gives us the first part of the theorem.
Upper bound. By considering the definition and equality of , we clearly have
and then
Hence, we get the righthand 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 .
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 in terms of the sum of the ith row of the Kirchhoff matrix 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 was labeled by in Section 2.)
Lemma 2LetGbe a connected graph withvertices. Then
whereis the sum of theith row of. Here, the equality holds if and only if.
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 connectedgraph with. Then we have
In (13) the equality holds if and only if.
Proof As a special case of the theory, if we assume that G is a null graph , then for each , we get and . Thus and equality holds in Equation (13). In the reverse part, if , then by AGMI, one can easily see that and hence .
As a general case, let us suppose that . Therefore . We then have
Now, by considering the function with its derivative , where , we easily conclude that f is an increasing function for . Hence, from (14) and by Lemma 2, we get
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 . Since and , we must have . Thus G is a connected graph. Moreover, from the equality of (16), we have . Since and , by Lemma 2, G is a complete graph .
The converse part is clear, i.e., the equality holds in (13) for the complete graph .
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 with respect to the Kirchhoff energy .
Theorem 7LetGbe as above. Then
and
Equality holds in (17) or (18) if and only if.
Proof By considering the proof of Theorem 5, we have
Moreover, by considering the Kirchhoff energy defined in (6), we get
which leads to (as in Theorem 5)
Hence we obtain the inequality in (17).
Another approximation to obtain an upper bound related to the relationship between and can be presented as follows
which implies
as claimed in (18). By a similar idea as in the previous results, the equality holds in (17) and (18) if and only if . □
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 obtained from the incidence matrix M of D by deleting the row . 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 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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