Isospectral Cayley graphs with even and odd spectrum

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 $\{\Gamma, \Gamma^+\}$.