Forbidden induced subgraphs for graphs and signed graphs with eigenvalues bounded from below

Abstract The smallest eigenvalue of a graph is the smallest eigenvalue of its adjacency matrix. We show that the family of graphs with smallest eigenvalue at least - can be defined by a finite set of forbidden induced subgraphs if and only if < ^*, where ^* = ^1/2 + ^-1/2 2. 01980, and is the unique real root of x³ = x + 1. This resolves a question raised by Bussemaker and Neumaier. As a byproduct, we find all the limit points of smallest eigenvalues of graphs, supplementing Hoffman’s work on those limit points in [-2, ). We also prove that the same conclusion about forbidden subgraph characterization holds for signed graphs. Our impetus for the study of signed graphs is to determine the maximum cardinality of a spherical two-distance set with two fixed angles (one acute and one obtuse) in high dimensions. Denote by N, (d) the maximum number of unit vectors in Rᵈ where all pairwise inner products lie in \, \ with -1 < 0 < 1. Very recently Jiang, Tidor, Yao, Zhang, and Zhao determined the limit of N, (d) /d as d when + 2 < 0 or (1-) / (-) \1, 2, 3\, and they proposed a conjecture on the limit in terms of eigenvalue multiplicities of signed graphs. We establish their conjecture whenever (1-) / (-) < ^*.