Abstract
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 $\tau$.
Highlights
Let Γ = (V (Γ), E(Γ)) be a simple undirected graph of order n with adjacency matrix A(Γ)
We first show that, for each i ∈ [n], the left coset decomposition of G with respect to the stabilizer subgroup Gi is an equitable partition of Γ, and all these equitable partitions share the same quotient matrix BΠ. We prove that those eigenvalues of Γ not belonging to BΠ can be bounded above by the sum of second eigenvalues of some subgraphs of Γ
By using Theorem 7, we reduce the problem of determining the second eigenvalues of normal Cayley graphs of highly transitive groups to that of verifying the result for some smaller graphs
Summary
In 1992, Aldous [1] (see [19, 9]) conjectured that the spectral gap of Cay(Sn, T ) is equal to the algebraic connectivity (second least Laplacian eigenvalue) of Tra(T ). Most of the above results rely heavily on the representation theory of the symmetric group Sn. The second eigenvalues of Cayley graphs of the symmetric group Sn or the alternating groups An have been determined for some special generators that are not transpositions. We determine the second eigenvalues of some subgraphs (over one hundred families) of these 41 families of normal Cayley graphs From these results we can determine the spectral gap of Cay(Sn, {(p, q) | 1 p, q n}) (previously done by Diaconis and Shahshahani [15]) and Cay(Sn, {(1, q) | 2 q n}) (previously obtained by Flatto, Odlyzko and Wales [18, Theorem 3.7]). We show that a recent conjecture of Dai [14] is true as a consequence of Aldous’ theorem and we discuss some related questions and open problems
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