A simple arithmetic criterion for graphs being determined by their generalized spectra

Wang, Wei

doi:10.1016/j.jctb.2016.07.004

Cited by 49 publications

(57 citation statements)

References 11 publications

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“…is odd and square-free then G is determined by its generalized spectrum [20]. This shows that the divisor 2 ⌊ n 2 ⌋ plays a special role for the determinant of the walk matrix, in some sense C is an extremal divisibility condition, see also Section 6.2 in [13].…”

Section: Introductionmentioning

confidence: 92%

“…The following theorem due to Wang [20] characterizes certain DGS graphs by an arithmetic property of the determinant of their walk matrix.…”

Section: Constructing Dgs Graphsmentioning

confidence: 99%

“…Theorem 3.2 (Wang [20]). Let G be a graph on n vertices with n ≥ 6 and walk matrix W. Then 2 ⌊ n 2 ⌋ divides det(W ) .…”

Section: Constructing Dgs Graphsmentioning

confidence: 99%

“…In addition all rose graphs are determined by their Laplacian spectrum except for two specific examples, see [12].In this paper we construct infinite sequences of DGS graphs from certain small starter graphs. This construction involves the walk matrix of a graph and recent results in [20]. Let n be the number of vertices of G and let e be the column vector of height n with all entries equal to 1.…”

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confidence: 99%

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New families of graphs determined by their generalized spectrum

Liu

Siemons

Wang

2019

Discrete Mathematics

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We construct infinite families of graphs that are determined by their generalized spectrum. This construction is based on new formulae for the determinant of the walk matrix of a graph. The graphs constructed here all satisfy a lower divisibility for the determinant of their walk matrix. 1 spectrum, or a DS graph for short. In [4,8,9] it is conjectured that almost all graphs are DS, more recent surveys can be found in [5].A variant of this problem concerns the generalized spectrum of G which is given by the pair spec(G), spec(G) . In this situation G is said to be determined by its generalized spectrum, or a DGS graph for short, if spec(H), spec(H) = spec(G), spec(G) implies that H is isomorphic to G. For the most recent results on DGS graphs see [13] and [20] in particular. To date only a few families of graphs are known to be DS or DGS. This includes some almost complete graphs [1], pineapple graphs [10] and kite graphs [11]. These are all known to be DS, and in particular DGS. In addition all rose graphs are determined by their Laplacian spectrum except for two specific examples, see [12].In this paper we construct infinite sequences of DGS graphs from certain small starter graphs. This construction involves the walk matrix of a graph and recent results in [20]. Let n be the number of vertices of G and let e be the column vector of height n with all entries equal to 1. Then the walk matrix of G is the n × n matrix W = e, Ae, A 2 e, · · · , A n−1 e

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Section: Introductionmentioning

confidence: 92%

“…The following theorem due to Wang [20] characterizes certain DGS graphs by an arithmetic property of the determinant of their walk matrix.…”

Section: Constructing Dgs Graphsmentioning

confidence: 99%

“…Theorem 3.2 (Wang [20]). Let G be a graph on n vertices with n ≥ 6 and walk matrix W. Then 2 ⌊ n 2 ⌋ divides det(W ) .…”

Section: Constructing Dgs Graphsmentioning

confidence: 99%

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confidence: 99%

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New families of graphs determined by their generalized spectrum

Liu

Siemons

Wang

2019

Discrete Mathematics

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“…Before controllable graphs (on n vertices) were given their nomenclature, they featured in the paper [23] as the class of graphs G n . In [22,23], it was proved that certain subclasses of G n are determined by their generalized spectrum. Other references do not focus on controllable graphs but still describe classes of graphs that are determined by their spectrum; for a survey of results on such graphs, the reader is referred to [20,21].…”

Section: Introductionmentioning

confidence: 99%

The Overgraphs of Generalized Cospectral Controllable Graphs

Farrugia¹

2019

Electron. J. Combin.

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Two graphs are said to be generalized cospectral if they have the same characteristic polynomials and so do their complements. A graph is controllable if its walk matrix is nonsingular; equivalently, if all the eigenvalues of its adjacency matrix are simple and main. A graph $H$ on $(n+1)$ vertices is an overgraph of another graph $G$ on $n$ vertices if $G$ is a vertex-deleted subgraph of $H$. We prove that no two distinct overgraphs of a controllable graph are generalized cospectral; this strengthens an earlier result that stated that no two such overgraphs are isomorphic. Moreover, we present methods that produce pairs of generalized cospectral graphs $G^\prime$ and $H^\prime$ starting from a pair of generalized cospectral, non-isomorphic, controllable graphs $G$ and $H$. We show that if $G^\prime$ and $H^\prime$ are controllable, then they are non-isomorphic.

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Unlocking the walk matrix of a graph

Liu

Siemons

2021

J Algebr Comb

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Let G be a graph of order n, with vertex set V = {v 1 , . . . , v n } and adjacency matrix A. For a non-empty set S of vertices let e = (x 1 , . . . , x n ) T be the characteristic vector of S, that is, x = 1 if v ∈ S and x = 0 otherwise. Then the n × n matrix W S := e, Ae, A 2 e, . . . , A n−1 e is the walk matrix of G for S. This term refers to the fact that in W S the kth entry in the row corresponding to v is the number of walks of length k − 1 from v to some vertex in S. Let μ 1 , . . . , μ s be the distinct eigenvalues of A. For given S and characteristic vector e, we re-arrange these eigenvalues in such a way that SD(S) : e = e 1 + e 2 + • • • + e r(1)for a certain r ≤ s, where the e i are eigenvectors of A for eigenvalue μ i , for all 1 ≤ i ≤ r . We refer to (1) as the spectral decomposition of S, or more properly, of its characteristic vector e. We show that the walk matrix W S determines the spectral decomposition of S and vice versa. Explicit algorithms are given which establish this correspondence. In particular, we show that the number r of distinct eigenvectors that appear in (1) is equal to the rank of W S . Various results can be derived from this theorem. We show that W S determines the adjacency matrix of G if W S has rank ≥ n − 1. Another application is that if W S has rank ≥ n − 1, then another graph G *

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A simple arithmetic criterion for graphs being determined by their generalized spectra

Cited by 49 publications

References 11 publications

New families of graphs determined by their generalized spectrum

New families of graphs determined by their generalized spectrum

The Overgraphs of Generalized Cospectral Controllable Graphs

Unlocking the walk matrix of a graph

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