The Second Eigenvalue of some Normal Cayley Graphs of Highly Transitive Groups

Xueyi Huang, Qiongxiang Huang, Sebastian M. Cioabă
2019 Electronic Journal of Combinatorics  
Let $G$ be a finite group acting transitively on $[n]=\{1,2,\ldots,n\}$, and let $\Gamma=\mathrm{Cay}(G,T)$ be a Cayley graph of $G$. The graph $\Gamma$ 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 $\Gamma$ in terms of the second eigenvalues of certain subgraphs of $\Gamma$. Using this result, we develop a recursive method to determine the second eigenvalues of certain Cayley
more » ... s 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 $\max_{\tau\in T}|\mathrm{supp}(\tau)|\leqslant 5$, where $\mathrm{supp}(\tau)$ is the set of points in $[n]$ non-fixed by $\tau$.
doi:10.37236/8054 fatcat:za27qw27pzcylccjkcnyfspt5q