A generalization of a theorem of Hoffman

Abstract

In 1977, Hoffman gave a characterization of graphs with smallest eigenvalue at least −2. In this paper we generalize this result to graphs with smaller smallest eigenvalue. For the proof, we use a combinatorial object named Hoffman graph, introduced by Woo and Neumaier in 1995. Our result says that for every λ ≤ −2, if a graph with smallest eigenvalue at least λ satisfies some local conditions, then it is highly structured. We apply our result to graphs which are cospectral with the Hamming graph H(3, q), the Johnson graph J(v, 3) and the 2clique extension of grids, respectively.

On graphs with smallest eigenvalue at least −3 and their lattices

“…Now we give definitions and preliminaries of Hoffman graphs. For more details, see [9], [12] and [19]. (i) Every vertex with label f is adjacent to at least one vertex with label s;…”

“…In this subsection, we summarize some facts about associated Hoffman graphs and quasi-cliques, which provide some connection between Hoffman graphs and graphs. For more details, we refer to [10] and [12].…”

An Application of Hoffman Graphs for Spectral Characterizations of Graphs

“…Regular graphs with four distinct eigenvalues have been previously studied [4], and a key observation that we will use is that these graphs are walk-regular, which implies strong combinatorial information on the graph. The starting point for our work is a result by Koolen et al [14]:…”

Problems on Graphs with Fixed Smallest Eigenvalue

“…They did not need the assumption of co-edge regularity, but they needed a very large lower bound on the valency. In their proof, they used the following result of Koolen et al [64]. A real number λ is an eigenvalue of (G, τ ), if λ is an eigenvalue of its adjacency matrix A(G, τ ).…”

“…[64, Theorem 1.4]). There exists a positive integer κ 2 such that, if a graph G satisfies the following conditions:…”