On strongly asymmetric and controllable primitive graphs

Farrugia, Alexander

doi:10.1016/j.dam.2016.04.001

Cited by 7 publications

(4 citation statements)

References 12 publications

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“…where b is a 0-1 vector. Usually, b is taken to be j, the vector of all ones [11][12][13][14], but there are exceptions [15][16][17][18]. For every i and j, the entry in the ith row and jth column of W b is equal to the number of walks of length j − 1 that start from vertex i and end at any vertex in S, where S is the subset of V (G) indicated by the entries in b that are equal to 1.…”

Section: Walk Matricesmentioning

confidence: 99%

Graphs Having Most of Their Eigenvalues Shared by a Vertex Deleted Subgraph

Farrugia

2021

Symmetry

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Let G be a simple graph and {1,2,…,n} be its vertex set. The polynomial reconstruction problem asks the question: given a deck P(G) containing the n characteristic polynomials of the vertex deleted subgraphs G−1, G−2, …, G−n of G, can ϕ(G,x), the characteristic polynomial of G, be reconstructed uniquely? To date, this long-standing problem has only been solved in the affirmative for some specific classes of graphs. We prove that if there exists a vertex v such that more than half of the eigenvalues of G are shared with those of G−v, then this fact is recognizable from P(G), which allows the reconstruction of ϕ(G,x). To accomplish this, we make use of determinants of certain walk matrices of G. Our main result is used, in particular, to prove that the reconstruction of the characteristic polynomial from P(G) is possible for a large subclass of disconnected graphs, strengthening a result by Sciriha and Formosa.

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Section: Walk Matricesmentioning

confidence: 99%

Graphs Having Most of Their Eigenvalues Shared by a Vertex Deleted Subgraph

Farrugia

2021

Symmetry

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“…Equivalently, as mentioned in the introduction, G is controllable if all of its eigenvalues are simple and main [5], which is the case if and only if G has only main eigenvectors [6]. Further results on controllable graphs were reported in [7,19].…”

Section: Preliminariesmentioning

confidence: 99%

“…In [7,Theorem 3.5], it was proved that all overgraphs of a controllable graph G are non-isomorphic. Thanks to Corollary 14, we can now prove an even stronger result, which is the second main result of this paper.…”

Section: Constructing Generalized Cospectral Overgraphs From Generalimentioning

confidence: 99%

“…We remark that, by Lemma 6 and Theorem 12, the overgraphs G + j and H + j are generalized cospectral if and only if G and H are generalized cospectral; moreover, they are non-isomorphic whenever G and H are non-isomorphic. Additionally, if G and H are controllable, then G + j and H + j are controllable if and only if 0 is not an eigenvalue of G [7].…”

Section: Constructing Generalized Cospectral Overgraphs From Generalimentioning

confidence: 99%

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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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Graphs with n−1 main eigenvalues

Liu

et al. 2021

Discrete Mathematics

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On strongly asymmetric and controllable primitive graphs

Cited by 7 publications

References 12 publications

Graphs Having Most of Their Eigenvalues Shared by a Vertex Deleted Subgraph

Graphs Having Most of Their Eigenvalues Shared by a Vertex Deleted Subgraph

The Overgraphs of Generalized Cospectral Controllable Graphs

Graphs with n−1 main eigenvalues

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