Isospectral Cayley graphs with even and odd spectrum
Paula M. Chiapparoli, Ricardo A. Podestá
TL;DR
The paper introduces mirror di-Cayley graphs (MDCGs) MX(G;S,T) and MX^+(G;S,T), then concentrates on the finite-family MX^*(G;S,T) with T in { {e}, S, S∪{e} }. It proves that MDCG spectra derive directly from the spectrum of the underlying Cayley graph X^*(G,S), yielding integrality transfer and a clear parity dichotomy: MX^*(G;S,S) has even spectrum while MX^*(G;S,S∪{e}) has odd spectrum when X^*(G,S) is integral. The authors provide explicit spectral formulas in terms of irreducible characters and establish isospectrality relations: MDCGs are isospectral exactly when their underlying Cayley graphs are, and they construct infinite families of integral isospectral Cayley graphs with both even and odd spectra using unitary Cayley graphs over finite rings and NEPS/Cayley-graph decompositions. The work delivers concrete examples (including rings with Artin decompositions) and demonstrates that integral even/odd Cayley graphs can be systematically generated via MDCGs, enriching the catalog of isospectral pairs with prescribed parity and integrality properties.
Abstract
For a group $G$ and subsets $S,T \subset G$ we introduce the mirror di-Cayley graph $MX(G;S,T)$ and mirror di-Cayley sum graph $MX^+(G;S,T)$ with connections sets $S$ and $T$ (MDCGs for short). We refer to them indistinctly by $MX^*(G;S,T)$. We then consider the family $\mathcal{F}$ of those MDCGs with $T \in \mathcal{S}$, where $\mathcal{S}= \big\{ \{e\}, S, S \cup \{e\} \big\}$. We compute the spectra of the graphs $MX^*(G;S,T)$, with $T \in \mathcal{S}$, in terms of those of the corresponding Cayley graphs $X^*(G,S)$. We show that if $X(G,S)$ has integral spectrum then $MX^*(G;S,T)$ is also integral for any $T \in \mathcal{S}$, but $MX^*(G;S,S)$ has even spectrum (all even eigenvalues) and $MX^*(G;S,S \cup \{e\})$ has odd spectrum (all odd eigenvalues), an interesting phenomenom which seems to be new. We then study isospectrality between different pairs of MDCGs in terms of the isospectrality of the underlying Cayley graphs. Finally, using unitary Cayley graphs $X(R,R^*)$ over a finite commutative ring $R$, which is known to be integral, we construct pairs of integral isospectral mirror di-Cayley (sum) graphs $\{ MX(R;R^*, T), MX^+(R;R^*, T) \}$, both with even (resp.\@ odd) spectrum for $T=R^*$ (resp.\@ $T=R^* \cup \{0\}$). All these examples can be seen as Cayley (sum) graphs over $G=R \times \mathbb{Z}_2$, hence obtaining pairs of even and odd isospectral Cayley graphs of the form $\{Γ, Γ^+\}$.