A note on the spectra and eigenspaces of the universal adjacency matrices of arbitrary lifts of graphs

TL;DR

This paper investigates the spectra and eigenspaces of universal adjacency matrices of arbitrary graph lifts, extending previous methods to a broader class of matrices including adjacency, Laplacian, and Seidel matrices.

Contribution

It generalizes existing spectral analysis techniques to all universal adjacency matrices of arbitrary graph lifts, regardless of regularity.

Findings

Spectra of universal adjacency matrices can be determined using a unified method.

Eigenspaces of these matrices are characterized for arbitrary lifts.

Applicable to various matrices like adjacency, Laplacian, and Seidel matrices.

Abstract

The universal adjacency matrix $U$ of a graph $\Gamma$, 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 $c_i\in {\mathbb R}$ and $c_1\neq 0$. Thus, as particular cases, $U$ may be the adjacency matrix, the Laplacian, the signless Laplacian, and the Seidel matrix. In this note, we show that basically the same method introduced before by the authors can be applied for determining the spectra and bases of all the corresponding eigenspaces of arbitrary lifts of graphs (regular or not).