The second largest eigenvalues of some Cayley graphs on alternating groups

TL;DR

This paper determines the second largest eigenvalues of specific Cayley graphs on alternating groups, providing insights into their spectral properties and structural characteristics.

Contribution

It explicitly computes the second largest eigenvalues for the Cayley graphs on alternating groups defined by particular generating sets, a novel spectral analysis in this context.

Findings

Second largest eigenvalues of $AG_n$, $EAG_n$, and $CAG_n$ are explicitly determined.

Spectral properties of these Cayley graphs are characterized.

Results contribute to understanding the expansion and connectivity of these graphs.

Abstract

Let $A_n$ denote the alternating group of degree $n$ with $n\geq 3$. The alternating group graph $AG_n$, extended alternating group graph $EAG_n$ and complete alternating group graph $CAG_n$ are the Cayley graphs $\mathrm{Cay}(A_n,T_1)$, $\mathrm{Cay}(A_n,T_2)$ and $\mathrm{Cay}(A_n,T_3)$, respectively, where $T_1=\{(1,2,i),(1,i,2)\mid 3\leq i\leq n\}$, $T_2=\{(1,i,j),(1,j,i)\mid 2\leq i<j\leq n\}$ and $T_3=\{(i,j,k),(i,k,j)\mid 1\leq i<j<k\leq n\}$. In this paper, we determine the second largest eigenvalues of $AG_n$, $EAG_n$ and $CAG_n$.