Geodesic
The second eigenvalue of some normal Cayley graphs of highly transitive groups
Xueyi Huang1 ; Qiongxiang Huang2 ; Sebastian M. Cioabă3
1Zhengzhou University
2Xinjiang University
3University of Delaware
The electronic journal of combinatorics, Tome 26 (2019) no. 2
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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 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
Classification : 05C50, 05C25
Mots-clés : connected normal Cayley graphs

Xueyi Huang  1   ; Qiongxiang Huang  2   ; Sebastian M. Cioabă  3

1 Zhengzhou University
2 Xinjiang University
3 University of Delaware 
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     author = {Xueyi Huang and Qiongxiang Huang and Sebastian M. Cioab\u{a}},
     title = {The second eigenvalue of some normal {Cayley} graphs of highly transitive groups},
     journal = {The electronic journal of combinatorics},
     year = {2019},
     volume = {26},
     number = {2},
     doi = {10.37236/8054},
     zbl = {1416.05174},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/8054/}
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Xueyi Huang; Qiongxiang Huang; Sebastian M. Cioabă. The second eigenvalue of some normal Cayley graphs of highly transitive groups. The electronic journal of combinatorics, Tome 26 (2019) no. 2. doi: 10.37236/8054

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