The minimum rank of universal adjacency matrices

Ahmadi, Bahman; Alinaghipour, Fatemeh; Fallat, Shaun M.; Fan, Yi-Zheng; Meagher, Karen; Nasserasr, Shahla

doi:10.1016/j.laa.2012.05.033

Cited by 3 publications

(6 citation statements)

References 5 publications

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“…, d n ), I is the identity matrix, and J is the all-1 matrix. See, for instance, Haemers and Omidi [15], Ahmadi, Alinaghipour, Fallat, Fan, Meagher, and Nasserasr [1], or Farrugia and Sciriha [10]. Thus, for given values of the coefficients (c 1 , c 2 , c 3 , c 4 ), the universal adjacency matrix particularizes to important matrices used in algebraic graph theory, such as the adjacency matrix (1, 0, 0, 0), the Laplacian (−1, 1, 0, 0), the signless Laplacian (1, 1, 0, 0), and the Seidel matrix (−2, 0, −1, 1).…”

Section: Introductionmentioning

confidence: 99%

On the spectra and eigenspaces of the universal adjacency matrices of arbitrary lifts of graphs

Dalfó

Fiol

Pavlíková

et al. 2022

Linear and Multilinear Algebra

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The universal adjacency matrix U of a graph Γ, with adjacency matrix A, is a linear combination of A, the diagonal matrix D of vertex degrees, the identity matrix I, and the all-1 matrix J with real coefficients, that is, U = c 1 A+c 2 D+c 3 I +c 4 J, with c i ∈ R and c 1 = 0. Thus, in particular cases, U may be the adjacency matrix, the Laplacian, the signless Laplacian, and the Seidel matrix. In this paper, we develop a method for determining the universal spectra and bases of all the corresponding eigenspaces of arbitrary lifts of graphs (regular or not). As an example, the method is applied to give an efficient algorithm to determine the characteristic polynomial of the Laplacian matrix of the symmetric squares of odd cycles, together with closed formulas for some of their eigenvalues.

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Section: Introductionmentioning

confidence: 99%

On the spectra and eigenspaces of the universal adjacency matrices of arbitrary lifts of graphs

Dalfó

Fiol

Pavlíková

et al. 2022

Linear and Multilinear Algebra

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“…. , v n are the vertices of G, then the matrix D is the diagonal matrix whose i th diagonal entry is the degree of v i for all 1 ≤ i ≤ n. The vector j denotes the vector 1…”

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confidence: 99%

“…In Section 2, we describe the universal adjacency matrix U [1,8] associated with a graph G and prove that its main eigenvalues must have an associated eigenvector of a certain form, from which an upper bound on the number of main eigenvalues of U is deduced. In Section 3, we summarize the principal results from control theory that will be used in subsequent sections, leading to the definition of a U-controllable graph G in Section 4, where several characterisations of such a graph are presented.…”

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confidence: 99%

“…The universal adjacency matrix U. The reference [8] (see also [1]) defined a universal adjacency matrix associated with a simple graph G to be a matrix of the form…”

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confidence: 99%

“…The authors of [8] determined the graphs that admit a universal adjacency matrix having only two distinct eigenvalues. In [1], the minimum rank of U was studied. In this article, we consider the number of distinct main eigenvalues of U and relate this number to what we call U-controllable graphs.…”

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confidence: 99%

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On the Main Eigenvalues of Universal Adjacency Matrices and U-Controllable Graphs

Farrugia

Sciriha

2015

ELA

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Abstract. A universal adjacency matrix U of a graph G is a linear combination of the 0-1 adjacency matrix A, the diagonal matrix of vertex degrees D, the identity matrix I and the matrix J each of whose entries is 1. A main eigenvalue of U is an eigenvalue having an eigenvector that is not orthogonal to the all-ones vector. It is shown that the number of distinct main eigenvalues of U associated with a simple graph G is at most the number of orbits of any automorphism of G. The definition of a U-controllable graph is given using control-theoretic techniques and several necessary and sufficient conditions for a graph to be U-controllable are determined. It is then demonstrated that U-controllable graphs are asymmetric and that the converse is false, showing that there exist both regular and non-regular asymmetric graphs that are not U-controllable for any universal adjacency matrix U. To aid in the discovery of these counterexamples, a γ-Laplacian matrix L (γ) is used, which is a simplified form of U. It is proved that any U-controllable graph is a L (γ)-controllable graph for some parameter γ.

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Perfect codes and universal adjacency spectra of commuting graphs of finite groups

Banerjee

2022

J. Algebra Appl.

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The commuting graph [Formula: see text] of a finite group [Formula: see text] has vertex set as [Formula: see text], and any two distinct vertices [Formula: see text] are adjacent if [Formula: see text] and [Formula: see text] commute with each other. In this paper, we first study the perfect codes of [Formula: see text]. We then find the universal adjacency spectra of the join of two regular graphs, join of two regular graphs in which one graph is a union of two regular graphs, and generalized join of regular graphs in terms of adjacency spectra of the constituent graphs and an auxiliary matrix. As a consequence, we obtain the adjacency, Laplacian, signless Laplacian, and Seidel spectra of the above graph operations. As an application of the results obtained, we calculate the adjacency, Laplacian, signless Laplacian, and Seidel spectra of [Formula: see text] for [Formula: see text], where [Formula: see text] is the dihedral group, [Formula: see text] is the dicyclic group and [Formula: see text] is the semidihedral group. Moreover, we provide the exact value of the spectral radius of the adjacency, Laplacian, signless Laplacian, and Seidel matrix of [Formula: see text] for [Formula: see text]. Some of the theorems published in [F. Ali and Y. Li, The connectivity and the spectral radius of commuting graphs on certain finite groups, Linear Multilinear Algebra 69 (2019) 1–14; T. Cheng, M. Dehmer, F. Emmert-Streib, Y. Li and W. Liu, Properties of commuting graphs over semidihedral groups, Symmetry 13(1) (2021) 103] can be deduced as corollaries from the theorems obtained in this paper.

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The minimum rank of universal adjacency matrices

Cited by 3 publications

References 5 publications

On the spectra and eigenspaces of the universal adjacency matrices of arbitrary lifts of graphs

On the spectra and eigenspaces of the universal adjacency matrices of arbitrary lifts of graphs

On the Main Eigenvalues of Universal Adjacency Matrices and U-Controllable Graphs

Perfect codes and universal adjacency spectra of commuting graphs of finite groups

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