Spectra and Laplacian spectra of arbitrary powers of lexicographic products of graphs

TL;DR

This paper derives the spectra and Laplacian spectra of lexicographic powers of graphs, providing formulas for regular and arbitrary graphs, and explores spectral properties of large graph powers with practical examples.

Contribution

It introduces explicit spectral and Laplacian spectral formulas for lexicographic powers of graphs, including regular and arbitrary cases, and extends spectral properties to these powers.

Findings

Spectra of lexicographic powers of regular graphs are explicitly determined.

Laplacian spectra of arbitrary graph powers are derived.

Spectral properties are extended to large graph powers, exemplified by the Petersen graph.

Abstract

Consider two graphs $G$ and $H$. Let $H^k[G]$ be the lexicographic product of $H^k$ and $G$, where $H^k$ is the lexicographic product of the graph $H$ by itself $k$ times. In this paper, we determine the spectrum of $H^k[G]$ and $H^k$ when $G$ and $H$ are regular and the Laplacian spectrum of $H^k[G]$ and $H^k$ for $G$ and $H$ arbitrary. Particular emphasis is given to the least eigenvalue of the adjacency matrix in the case of lexicographic powers of regular graphs, and to the algebraic connectivity and the largest Laplacian eigenvalues in the case of lexicographic powers of arbitrary graphs. This approach allows the determination of the spectrum (in case of regular graphs) and Laplacian spectrum (for arbitrary graphs) of huge graphs. As an example, the spectrum of the lexicographic power of the Petersen graph with the googol number (that is, $10^{100}$) of vertices is determined. The paper finish with the extension of some well known spectral and combinatorial invariant properties of graphs to its lexicographic powers.