General inertia bound for graphs

Prove that every graph G satisfies 2n^+(G) ≤ n^-(G)(n^-(G)+1), where n^+(G) and n^-(G) are the numbers of positive and negative adjacency-matrix eigenvalues, respectively, thereby extending the absolute bound for strongly regular graphs to all graphs.

Background

The paper’s principal conjecture proposes extending the classical absolute bound for strongly regular graphs to arbitrary graphs. The authors verify it for several classes, including planar graphs, line graphs, cographs, graphs of maximum degree four, and other families, and report computational verification for graphs of small order. The conjecture is also connected to upper-bounding positive inertia in terms of negative inertia and to Torgašev’s problem on reduced graphs.

References

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.

— New conjectures on the inertia of graphs  (2508.01163 - Akbari et al., 2 Aug 2025) in Abstract; Conjecture 1 in Section 1