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arXiv · 2609.30327

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

Paula M. Chiapparoli·Ricardo A. Podestá
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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 $MX^+(G;S,T)$, namely those with $T=\{e\}, S$ or $S \cup \{e\}$. We refer to them indistinctly by $MX^*(G;S,T)$. We can think of $MX^*(G;S,T)$ as two copies of the Cayley (sum) graph $X^*(G,S)$ 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 $MX^*(G;S,T)$ 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.

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Paula M. Chiapparoli, Ricardo A. Podestá. 2026-09-23. The energy of a family of mirror di-Cayley (sum) graphs: equienergy and moments. https://arxiv.org/abs/2609.30327

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