Conjectured bound for the distribution of eigenvalues of a graph

Abstract

Let $(n^+, n^0, n^-)$ denote the inertia of a graph $G$ with $n$ vertices. Nordhaus-Gaddum bounds are known for inertia, except for an upper bound for $n^-$. We conjecture that for any graph \[ n^-(G) + n^-(\bar{G}) \le 1.5(n - 1), \] and prove this bound for various classes of graphs and for almost all graphs. We consider the relationship between this bound and the number of eigenvalues that lie within the interval $-1$ to $0$, which we denote $n_{(-1,0)}(G)$. We conjecture that for any graph \[ n_{(-1,0)}(G) \le 0.5(n - 1). \] and prove this bound for almost all graphs. We also investigate extremal graphs for both bounds and show that both bounds are equivalent for regular graphs.