Conjectured bound for the distribution of eigenvalues of a graph

TL;DR

This paper proposes conjectured bounds on the distribution of eigenvalues related to a graph's inertia and eigenvalues in a specific interval, providing proofs for various graph classes and exploring extremal cases.

Contribution

It introduces new conjectured bounds on the inertia and eigenvalue distribution of graphs, with proofs for specific classes and analysis of extremal graphs.

Findings

Proved the bounds for various classes of graphs.

Established the bounds hold for almost all graphs.

Showed the bounds are equivalent for regular graphs.

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.