Lexicographic products and lexicographic powers of graphs -- a walk matrix approach

TL;DR

This paper explicitly determines the spectrum of lexicographic graph products and powers using a walk matrix approach, revealing conditions for eigenvalue relationships based on graph nullity and main eigenvalues.

Contribution

It introduces a walk matrix method to analyze spectra of lexicographic graph products and powers, linking eigenvalues of component graphs with the combined structure.

Findings

Explicit formulas for spectra of lexicographic products and powers.

Conditions for main eigenvalues to be preserved or altered.

Analysis of eigenvalue multiplicities related to graph nullity.

Abstract

The characteristic polynomial and the spectrum of the lexicographic product of graphs $H[G]$, a specific instance of the generalized composition (also called $H$-join), are explicitly determined for arbitrary graphs $H$ and $G$, in terms of the eigenvalues of $G$ and an $H[G]$ associated matrix $\widetilde{{\bf W}}$, which relates $H$ with $G$. This study also establishes conditions under which a main eigenvalue of $G$ is a main or non-main eigenvalue of the matrix $\widetilde{{\bf W}}$, when the nullity of the graph $H$ is $\eta>0$. In such a case, we prove that every main eigenvalue of $G$ is an eigenvalue of $\widetilde{{\bf W}}$ with multiplicity at least $\eta$ which is non-main for $\bf \widetilde{W}$ if and only if $0$ is a non-main eigenvalue of $H$. Furthermore, the spectra of the lexicographic powers of arbitrary graphs $G$ are analysed by applying the obtained results.