The second eigenvalue of some normal Cayley graphs of high transitive groups

TL;DR

This paper establishes bounds and recursive methods for determining the second eigenvalue of normal Cayley graphs of high transitive groups, especially symmetric groups, with applications to graphs with small support size.

Contribution

It provides a new upper bound for the second eigenvalue of normal Cayley graphs and a recursive approach to compute these eigenvalues for certain symmetric group graphs.

Findings

Derived an upper bound for the second eigenvalue in terms of subgraph eigenvalues.

Developed a recursive method for eigenvalue computation of Cayley graphs of $S_n$.

Determined second eigenvalues for most connected normal Cayley graphs with support size up to 5.

Abstract

Let $\Gamma$ be a finite group acting transitively on $[n]=\{1,2,\ldots,n\}$, and let $G=\mathrm{Cay}(\Gamma,T)$ be a Cayley graph of $\Gamma$. The graph $G$ is called normal if $T$ is closed under conjugation. In this paper, we obtain an upper bound for the second (largest) eigenvalue of the adjacency matrix of the graph $G$ in terms of the second eigenvalues of certain subgraphs of $G$ (see Theorem 2.6). Using this result, we develop a recursive method to determine the second eigenvalues of certain Cayley graphs of $S_n$ and we determine the second eigenvalues of a majority of the connected normal Cayley graphs (and some of their subgraphs) of $S_n$ with $\max_{\tau\in T}|\mathrm{supp}(\tau)|\leq 5$, where $\mathrm{supp}(\tau)$ is the set of points in $[n]$ non-fixed by $\tau$.