A complete characterization of graphs with second largest eigenvalue at most $(\sqrt5-1)/2$

Abstract

In 1993, Cao and Hong posed the problem of characterizing graphs whose second largest adjacency eigenvalue is greater than $1/3$ and less than $(\sqrt5-1)/2$. Subsequent work extended the question to the non-strict bound. We give a complete structural characterization of connected graphs $G$ with $|V(G)|\ge2$ satisfying $λ_2(G)\le(\sqrt5-1)/2$. Their complements are described by disjoint unions of graphs from an explicit family with least adjacency eigenvalue at least $-(1+\sqrt5)/2$, possibly together with a cone or a thin or thick spider constructed from disjoint unions of members of the same family, subject to explicit numerical inequalities. This provides a positive characterization of the class, going beyond the classical result of Cvetković and Simić that the bound $λ_2\le(\sqrt5-1)/2$ admits only finitely many minimal forbidden induced subgraphs.