# Universal Spectra of the Disjoint Union of Regular Graphs

Willem H. Haemers, Mohammad Reza Oboudi

Research output: Working paperOther research output

## Abstract

A universal adjacency matrix of a graph $G$ with adjacency matrix $A$ is any matrix of the form $U = \alpha A + \beta I + \gamma J + \delta D$ with $\alpha \neq 0$, where $I$ is the identity matrix, $J$ is the all-ones matrix and $D$ is the diagonal matrix with the vertex degrees. In the case that $G$ is the disjoint union of regular graphs, we present an expression for the characteristic polynomials of the various universal adjacency matrices in terms of the characteristic polynomials of the adjacency matrices of the components. As a consequence we obtain a formula for the characteristic polynomial of the Seidel matrix of $G$, and the signless Laplacian of the complement of $G$ (i.e. the join of regular graphs).
Original language English Ithaca Cornell University Library 4 Published - 6 Apr 2020

### Publication series

Name arXiv 2004.02499

• math.CO
• 05C50