Constructing cospectral graphs

Godsil, Chris; McKay, Brendan D.

doi:10.1007/bf02189621

Cited by 226 publications

(158 citation statements)

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“…In the above example, as it was mentioned in [11], if H has 2m vertices and a trivial automorphism group, than all 2m m possible realisations of H are non-isomorphic. By Theorem 4 the graphs G and G (π) are cospectral.…”

Section: Theorem 4 ( [11]mentioning

confidence: 83%

Graphs with Equal Irregularity Indices

Dimitrov

2014

APH

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The irregularity of a graph can be defined by different so-called graph topological indices. In this paper, we consider the irregularities of graphs with respect to the Collatz-Sinogowitz index [8], the variance of the vertex degrees [6], the irregularity of a graph [4], and the total irregularity of a graph [1]. It is known that these irregularity measures are not always compatible. Here, we investigate the problem of determining pairs or classes of graphs for which two or more of the above mentioned irregularity measures are equal. While in [17] this problem was tackled in the case of bidegreed graphs, here we go a step further by considering tridegreed graphs and graphs with arbitrarily large degree sets. In addition we pressent the smallest graphs for which all above irregularity indices are equal.

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Section: Theorem 4 ( [11]mentioning

confidence: 83%

Graphs with Equal Irregularity Indices

Dimitrov

2014

APH

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“…Proof Apply Godsil-McKay switching (cf. [3,7], 1.8.3, 14.2.3). Switch with respect to a 4-clique B such that every vertex outside B is adjacent to 0, 2 or 4 vertices inside.…”

Section: Cospectral Matesmentioning

confidence: 99%

Notes on simplicial rook graphs

et al. 2015

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The simplicial rook graph SR(m, n) is the graph of which the vertices are the sequences of nonnegative integers of length m summing to n, where two such sequences are adjacent when they differ in precisely two places. We show that SR(m, n) has integral eigenvalues, and smallest eigenvalue s = max −n, − m 2 , and that this graph has a large part of its spectrum in common with the Johnson graph J (m + n − 1, n). We determine the automorphism group and several other properties.

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“…Previously, Godsil and McKay [8] and more recently Haemers and Spence [10] have shown that the laplacian matrix has more representational power than the adjacency matrix, in terms of resulting in less number of cospectral graphs. According to the results given in [10], of more than a billion graphs with 11 vertices characterized by the adjacency matrix, approximately 21% is cospectral, while this fraction is only 9% for the laplacian matrix.…”

Section: Definitions and Preliminariesmentioning

confidence: 99%