Generalized Paley graphs equienergetic with their complements

TL;DR

This paper analyzes the spectra of generalized Paley graphs and their complements, establishing conditions for them to be equienergetic, and constructs infinite pairs of non-isospectral, regular, non-bipartite, non-strongly regular graphs.

Contribution

It computes spectra of specific generalized Paley graphs, derives recursive spectral formulas, and characterizes when these graphs and their complements are equienergetic, including new infinite graph pairs.

Findings

Spectra of $ ext{Paley}(3,q)$ and $ ext{Paley}(4,q)$ are explicitly computed.

Recursive formulas for spectra in non-semiprimitive cases are established.

Infinite pairs of equienergetic, non-isospectral, regular graphs are constructed.

Abstract

We consider generalized Paley graphs $\Gamma(k,q)$, generalized Paley sum graphs $\Gamma^+(k,q)$, and their corresponding complements $\bar \Gamma(k,q)$ and $\bar \Gamma^+(k,q)$, for $k=3,4$. Denote by $\Gamma = \Gamma^*(k,q)$ either $\Gamma(k,q)$ or $\Gamma^+(k,q)$. We compute the spectra of $\Gamma(3,q)$ and $\Gamma(4,q)$ and from them we obtain the spectra of $\Gamma^+(3,q)$ and $\Gamma^+(4,q)$ also. Then we show that, in the non-semiprimitive case, the spectrum of $\Gamma(3,p^{3\ell})$ and $\Gamma(4,p^{4\ell})$ with $p$ prime can be recursively obtained, under certain arithmetic conditions, from the spectrum of the graphs $\Gamma(3,p)$ and $\Gamma(4,p)$ for any $\ell \in \mathbb{N}$, respectively. Using the spectra of these graphs we give necessary and sufficient conditions on the spectrum of $\Gamma^*(k,q)$ such that $\Gamma^*(k,q)$ and $\bar \Gamma^*(k,q)$ are equienergetic for $k=3,4$. In a previous work we have classified all bipartite regular graphs $\Gamma_{bip}$ and all strongly regular graphs $\Gamma_{srg}$ which are complementary equienergetic, i.e.\@ $\{\Gamma_{bip}, \bar{\Gamma}_{bip}\}$ and $\{\Gamma_{srg}, \bar{\Gamma}_{srg}\}$ are equienergetic pairs of graphs. Here we construct infinite pairs of equienergetic non-isospectral regular graphs $\{\Gamma, \bar \Gamma\}$ which are neither bipartite nor strongly regular.