A new conjecture on the inertia of graphs

TL;DR

This paper proposes a new conjecture relating the positive and negative eigenvalues of a graph's adjacency matrix, extends it to special graph families, and explores related spectral properties of line graphs.

Contribution

It introduces a novel conjecture on graph inertia, proves it for certain graph classes, and investigates related spectral inequalities for line graphs.

Findings

Conjecture holds for line graphs and planar graphs.

Examples where the conjecture is tight are provided.

Partial results support the conjecture for connected graphs.

Abstract

Let $G$ be a graph with adjacency matrix $A(G)$. We conjecture that \[2n^+(G) \le n^-(G)(n^-(G) + 1),\] where $n^+(G)$ and $n^-(G)$ denote the number of positive and negative eigenvalues of $A(G)$, respectively. This conjecture generalizes to all graphs the well-known absolute bound for strongly regular graphs. The conjecture also relates to a question posed by Torga\v{s}ev. We prove the conjecture for special graph families, including line graphs and planar graphs, and provide examples where the conjecture is exact. We also conjecture that for any connected graph $G$, its line graph $L(G)$ satisfies $n^+(L(G)) \le n^-(L(G)) + 1$, and obtain partial results.