Retracted Article: On the Kirchhoff matrix, a new Kirchhoff index and the Kirchhoff energy
© Maden et al.; licensee Springer 2013
Received: 4 March 2013
Accepted: 9 July 2013
Published: 24 July 2013
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.
KeywordsKirchhoff 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 . As depicted and studied in , the term ‘resistance distance’ was used for chemical and physical interpretation.
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.
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 , .
where each is as given in (2).
we can obtain a new lower and upper bound (see (5)) for this new Kirchhoff index as in the following proposition.
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.
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 , pertaining to the Laplacian matrix, can be thought of as the first . We note that the theory of energy-like graph invariants was firstly introduced by Consonni and Todeschini in . Later on, Nikiforov  extended the definition of energy to arbitrary matrices making thus possible to conceive the incidence energy  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.
where each (with ordering ) denotes the eigenvalues of the Kirchhoff matrix . 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.
where κ is the sum of the squares of entries of the Kirchhoff matrix .
Hence the result. □
The following lemma is needed for our other results that are given in this paper.
as required. □
Therefore this gives the upper bound for .
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.
Here, since , we have .
Hence the result. □
In the following, we present a new lower bound which is better than the lower bound given in Theorem 2.
Proof In the light of Theorem 2, if we show the validity of the lower bound, then this finishes the proof.
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.
By , 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
where are the K-eigenvalues of G.
Thus the main result of the subsection is the following.
Equality holds on both sides if and only if .
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.
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 .
Hence the result. □
In , by considering the maximum eigenvalue, Zhou et al. presented a lower bound for the reciprocal distance matrix in terms of the sum of the i th 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 i th 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 . (At this point we recall that for simplicity, each was labeled by in Section 2.)
where is the sum of the ith 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.
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 .
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 .
Equality holds in (17) or (18) if and only if .
Hence we obtain the inequality in (17).
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 [, 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 , 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.
This work was presented in The International Conference on the Theory, Methods and Applications of Nonlinear Equations.
- Xiao W, Gutman I: Resistance distance and Laplacian spectrum. Theor. Chem. Acc. 2003, 110: 284–289. 10.1007/s00214-003-0460-4View ArticleGoogle Scholar
- Xiao W, Gutman I: On resistance matrices. MATCH Commun. Math. Comput. Chem. 2003, 49: 67–81.MathSciNetMATHGoogle Scholar
- Klein DJ: Graph geometry, graph metrics & Wiener. Fifty years of the Wiener index. MATCH Commun. Math. Comput. Chem. 1997, 35: 7–27.MATHGoogle Scholar
- Chen H, Zhang F: Resistance distance and the normalized Laplacian spectrum. Discrete Appl. Math. 2007, 155: 654–661. 10.1016/j.dam.2006.09.008MathSciNetView ArticleMATHGoogle Scholar
- Buckley F, Harary F: Distance in Graphs. Addision-Wesley, Redwood; 1990.MATHGoogle Scholar
- Klein DJ, Randić M: Resistance distance, applied graph theory and discrete mathematics in chemistry (Saskatoon, SK, 1991). J. Math. Chem. 1993, 12(1–4):81–95.MathSciNetView ArticleGoogle Scholar
- Güngör AD, Cevik AS, Das KC: On the Kirchhoff index and the resistance-distance energy of a graph. MATCH Commun. Math. Comput. Chem. 2012, 67(2):541–556.MathSciNetGoogle Scholar
- Cvetković D, Rowlinson P, Simić SK: Introduction to the Theory of Graph Spectra. Cambridge University Press, Cambridge; 2010.MATHGoogle Scholar
- Graovac A, Gutman I, Trinajstić N: Topological Approach to the Chemistry of Conjugated Molecules. Springer, Berlin; 1977.View ArticleMATHGoogle Scholar
- Cvetković DM, Doob M, Sachs H: Spectra of Graphs - Theory and Application. Academic Press, New York; 1980.MATHGoogle Scholar
- Güngör AD, Cevik AS: On the Harary energy and Harary Estrada index of a graph. MATCH Commun. Math. Comput. Chem. 2010, 64(1):281–296.MathSciNetMATHGoogle Scholar
- Gutman I: The energy of a graph. Ber. Math.-Stat. Sekt. Forsch. Graz 1978, 103: 1–22.MathSciNetMATHGoogle Scholar
- Gutman I: The energy of a graph: old and new results. In Algebraic Combinatorics and Applications. Edited by: Betten A, Kohnert A, Laue R, Wassermann A. Springer, Berlin; 2001:196–211.View ArticleGoogle Scholar
- Zhou B, Gutman I: On Laplacian energy of graphs. MATCH Commun. Math. Comput. Chem. 2007, 57: 211–220.MathSciNetMATHGoogle Scholar
- Consonni V, Todeschini R: New spectral indices for molecule description. MATCH Commun. Math. Comput. Chem. 2008, 60: 3–14.MathSciNetMATHGoogle Scholar
- Nikiforov V: The energy of graphs and matrices. J. Math. Anal. Appl. 2007, 326: 1472–1475. 10.1016/j.jmaa.2006.03.072MathSciNetView ArticleMATHGoogle Scholar
- Jooyandeh MR, Kiani D, Mirzakkah M: Incidence energy of a graph. MATCH Commun. Math. Comput. Chem. 2009, 62: 561–572.MathSciNetMATHGoogle Scholar
- Koolen J, Moulton V: Maximal energy of graphs. Adv. Appl. Math. 2001, 26: 47–52. 10.1006/aama.2000.0705MathSciNetView ArticleMATHGoogle Scholar
- Koolen J, Moulton V: Maximal energy of bipartite graphs. Graphs Comb. 2003, 19: 131–135. 10.1007/s00373-002-0487-7MathSciNetView ArticleMATHGoogle Scholar
- Estrada E: Characterization of 3D molecular structure. Chem. Phys. Lett. 2000, 319: 713–718. 10.1016/S0009-2614(00)00158-5View ArticleGoogle Scholar
- Estrada E: Characterization of the folding degree of proteins. Bioinformatics 2002, 18: 697–704. 10.1093/bioinformatics/18.5.697View ArticleGoogle Scholar
- Estrada E: Characterization of amino acid contribution to the folding degree of proteins. Proteins 2004, 54: 727–737. 10.1002/prot.10609View ArticleGoogle Scholar
- Estrada E, Rodríguez-Velázguez JA: Subgraph centrality in complex networks. Phys. Rev. E 2005., 71: Article ID 056103Google Scholar
- Estrada E, Rodríguez-Velázguez JA: Spectral measures of bipartivity in complex networks. Phys. Rev. E 2005., 72: Article ID 046105Google Scholar
- Estrada E, Rodríguez-Velázguez JA, Randić M: Atomic branching in molecules. Int. J. Quant. Chem. 2006, 106: 823–832. 10.1002/qua.20850View ArticleGoogle Scholar
- Güngör AD, Bozkurt SB: On the distance Estrada index of graphs. Hacet. J. Math. Stat. 2009, 38(3):277–283.MathSciNetMATHGoogle Scholar
- Deng H, Radenković S, Gutman S: The Estrada index. In Applications of Graph Spectra. Edited by: Cvetković D, Gutman I. Math. Inst., Belgrade; 2009:123–140.Google Scholar
- Peňa JAD, Gutman I, Rada J: Estimating the Estrada index. Linear Algebra Appl. 2007, 427: 70–76. 10.1016/j.laa.2007.06.020MathSciNetView ArticleMATHGoogle Scholar
- Zhou B, Trinajstic N: Maximum eigenvalues of the reciprocal distance matrix and the reserve Wiener matrix. Int. J. Quant. Chem. 2008, 108: 858–864. 10.1002/qua.21558View ArticleGoogle Scholar
- Bondy JA, Murty USR Graduate Texts in Mathematics 244. In Graph Theory. Springer, New York; 2008.View ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. 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.