New conjectures on the inertia of 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.