Resistance distance and Kirchhoff index in circulant graphs

Zhang, Heping; Yang, Yujun

doi:10.1002/qua.21068

Cited by 114 publications

(56 citation statements)

References 30 publications

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“…Although this is the most common distance metric for graphs, other measures of distances between vertices are possible. Among those is the “resistance distance,” which is of more recent time,7 in which distance is measured by electric resistance assuming each edge carrying unit resistance, received some attention 21–32. This metric is based on laws governing current flow in electric networks of Kirchhoff, whose work is considered as one of the most important early contribution to Graph Theory 33…”

Section: Matrices Of Interest In Structure‐property‐activity Studiesmentioning

confidence: 99%

Novel graph distance matrix

et al. 2010

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We have introduced novel distance matrix for graphs, which is based on interpretation of columns of the adjacency matrix of a graph as a set of points in n-dimensional space, n being the number of vertices in the graph. Numerical values for the distances are based on the Euclidean distance between n points in n-dimensional space. In this way, we have combined the traditional representation of graphs (drawn as 2D object of no fixed geometry) with their representation in n-dimensional space, defined by a set of n-points that lead to a representation of definite geometry. The novel distance matrix, referred to as natural distance matrix, shows some structural properties and offers novel graph invariants as molecular descriptors for structure-property-activity studies. One of the novel graph descriptors is the modified connectivity index in which the bond contribution for (m, n) bond-type is given by 1/ radical(m + n), where m and n are the valence of the end vertices of the bond. The novel distance matrix (ND) can be reduced to sparse distance-adjacency matrix (DA), which can be viewed as specially weighted adjacency matrix of a graph. The quotient of the leading eigenvalues of novel distance-adjacency matrix and novel distance matrix, as illustrated on a collection of graphs of chemical interest, show parallelism with a simple measure of graph density, based on the quotient of the number of edges in a graph and the maximal possible number of edges for graphs of the same size.

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Section: Matrices Of Interest In Structure‐property‐activity Studiesmentioning

confidence: 99%

Novel graph distance matrix

et al. 2010

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“…In addition, we find the expression for the Foster's n-th formula.production. The effective resistance and the Kirchhoff index have been computed for some classes of graphs with symmetries, see [3,5,6]. In particular, Palacios in [7] gave a closed-form formula for the Kirchhoff index for distance-regular graphs and a class of graphs of diameter two.…”

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confidence: 99%

A formula for the Kirchhoff index

Bendito

Carmona

Encinas

et al. 2008

Int J of Quantum Chemistry

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ABSTRACT:We show here that the Kirchhoff index of a network is the average of the Wiener capacities of its vertices. Moreover, we obtain a closed-form formula for the effective resistance between any pair of vertices when the considered network has some symmetries, which allows us to give the corresponding formulas for the Kirchhoff index. In addition, we find the expression for the Foster's n-th formula.

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“…It is difficult to give closed-form formulae for general graphs [1, 4 -7]. But the Kirchhoff index has been given for some classes of graphs, such as cycles [8,9], complete graphs [9], geodetic graphs [7], distance-transitive graphs [7], circulant graphs [10], linear hexagonal chains [11], and so on [4,7,12,13]. In this paper, we devote ourselves to cyclopolyacenes.…”

Section: Introductionmentioning

confidence: 99%

Kirchhoff Index of Cyclopolyacenes

Wang¹,

Zhang

2010

Zeitschrift Für Naturforschung A

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The resistance distance between two vertices of a connected graph G is computed as the effective resistance between them in the corresponding network constructed from G by replacing each edge with a unit resistor. The Kirchhoff index of G is the sum of resistance distances between all pairs of vertices. In this paper, following the method of Y. J. Yang and H. P. Zhang in the proof of the Kirchhoff index of the linear hexagonal chain, we obtain the Kirchhoff index of cyclopolyacenes, denoted by HR n , in terms of its Laplacian spectrum. We show that the Kirchhoff index of HR n is approximately one third of its Wiener index.

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Resistance distance and Kirchhoff index in circulant graphs

Cited by 114 publications

References 30 publications

Novel graph distance matrix

Novel graph distance matrix

A formula for the Kirchhoff index

Kirchhoff Index of Cyclopolyacenes

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