arXiv subjects
Publications and source records attributed to Jason R. Vermette.
From Alon and Boppana, and Serre, we know that for any given integer $k\geq 3$ and real number $λ<2\sqrt{k-1}$, there are finitely many $k$-regular graphs whose second largest eigenvalue is at most $λ$. In this paper, we investigate the largest number of vertices of such graphs.
We determine all graphs for which the adjacency matrix has at most two eigenvalues (multiplicities included) not equal to $-2$, or $0$, and determine which of these graphs are determined by their adjacency spectrum.
The simplicial rook graph ${\rm SR}(m,n)$ is the graph of which the vertices are the sequences of nonnegative integers of length $m$ summing to $n$, where two such sequences are adjacent when they differ in precisely two places. We show that ${\rm SR}(m,n)$ has integral eigenvalues, and smallest eigenvalue $s = \max (-n, -{m \choose 2})$, and that this graph has a large part of its spectrum in common with the Johnson graph $J(m+n-1,n)$. We determine the automorphism group and several other properties.