The energy of a family of mirror di-Cayley (sum) graphs: equienergy and moments

Abstract

For a group $G$ and subsets $S,T\subset G$ we consider a family of mirror di-Cayley graphs $MX(G;S,T)$ and mirror di-Cayley sums graphs $M{X}^{+}(G;S,T)$, namely those with $T=\{e\},S$ or $S\cup \{e\}$. We refer to them indistinctly by $M{X}^{\ast }(G;S,T)$. We can think of $M{X}^{\ast }(G;S,T)$ as two copies of the Cayley (sum) graph $X^{\ast }(G,S)$ joined by edges determined by the connection set $T$. Recently, in the work \textit{Isospectral Cayley graphs with even and odd spectrum}, we study the spectrum of these graphs and several isospectrality problems. Here, we compute the energy and spectral $k$-moments of $M{X}^{\ast }(G;S,T)$ in terms of those of the underlying Cayley graphs $X(G,S)$. Then, we study energetic problems like hypo-, order-, and hyper-energeticity of these graphs. Finally, we give conditions for the existence of equienergetic pairs of non-isomorphic MDCGs.

arXiv comment25 pages