# Universal adjacency matrices with two eigenvalues

W.H. Haemers, G.R. Omidi

Research output: Contribution to journalArticleScientificpeer-review

### Abstract

Consider a graph Γ on n vertices with adjacency matrix A and degree sequence (d1,…,dn). A universal adjacency matrix of Γ is any matrix in Span {A,D,I,J} with a nonzero coefficient for A, where D=diag(d1,…,dn) and I and J are the n × n identity and all-ones matrix, respectively. Thus a universal adjacency matrix is a common generalization of the adjacency, the Laplacian, the signless Laplacian and the Seidel matrix. We investigate graphs for which some universal adjacency matrix has just two eigenvalues. The regular ones are strongly regular, complete or empty, but several other interesting classes occur.
Original language English 2520-2529 Linear Algebra and its Applications 435 10 https://doi.org/10.1016/j.laa.2011.02.023 Published - 2011

### Fingerprint

Eigenvalue
Signless Laplacian
Degree Sequence
Graph in graph theory
Coefficient

### Cite this

@article{cd0dbc00625e413fa7a6ac72c5aa9a5f,
title = "Universal adjacency matrices with two eigenvalues",
abstract = "Consider a graph Γ on n vertices with adjacency matrix A and degree sequence (d1,…,dn). A universal adjacency matrix of Γ is any matrix in Span {A,D,I,J} with a nonzero coefficient for A, where D=diag(d1,…,dn) and I and J are the n × n identity and all-ones matrix, respectively. Thus a universal adjacency matrix is a common generalization of the adjacency, the Laplacian, the signless Laplacian and the Seidel matrix. We investigate graphs for which some universal adjacency matrix has just two eigenvalues. The regular ones are strongly regular, complete or empty, but several other interesting classes occur.",
author = "W.H. Haemers and G.R. Omidi",
note = "Appeared previously as CentER Discussion Paper 2010-119",
year = "2011",
doi = "10.1016/j.laa.2011.02.023",
language = "English",
volume = "435",
pages = "2520--2529",
journal = "Linear Algebra and its Applications",
issn = "0024-3795",
publisher = "Elsevier Inc.",
number = "10",

}

In: Linear Algebra and its Applications, Vol. 435, No. 10, 2011, p. 2520-2529.

Research output: Contribution to journalArticleScientificpeer-review

TY - JOUR

T1 - Universal adjacency matrices with two eigenvalues

AU - Haemers, W.H.

AU - Omidi, G.R.

N1 - Appeared previously as CentER Discussion Paper 2010-119

PY - 2011

Y1 - 2011

N2 - Consider a graph Γ on n vertices with adjacency matrix A and degree sequence (d1,…,dn). A universal adjacency matrix of Γ is any matrix in Span {A,D,I,J} with a nonzero coefficient for A, where D=diag(d1,…,dn) and I and J are the n × n identity and all-ones matrix, respectively. Thus a universal adjacency matrix is a common generalization of the adjacency, the Laplacian, the signless Laplacian and the Seidel matrix. We investigate graphs for which some universal adjacency matrix has just two eigenvalues. The regular ones are strongly regular, complete or empty, but several other interesting classes occur.

AB - Consider a graph Γ on n vertices with adjacency matrix A and degree sequence (d1,…,dn). A universal adjacency matrix of Γ is any matrix in Span {A,D,I,J} with a nonzero coefficient for A, where D=diag(d1,…,dn) and I and J are the n × n identity and all-ones matrix, respectively. Thus a universal adjacency matrix is a common generalization of the adjacency, the Laplacian, the signless Laplacian and the Seidel matrix. We investigate graphs for which some universal adjacency matrix has just two eigenvalues. The regular ones are strongly regular, complete or empty, but several other interesting classes occur.

U2 - 10.1016/j.laa.2011.02.023

DO - 10.1016/j.laa.2011.02.023

M3 - Article

VL - 435

SP - 2520

EP - 2529

JO - Linear Algebra and its Applications

JF - Linear Algebra and its Applications

SN - 0024-3795

IS - 10

ER -