Conjectured bounds for the sum of squares of positive eigenvalues of a graph

TL;DR

This paper discusses a conjecture about bounds on the sum of squares of positive eigenvalues in graphs, providing proofs for specific graph classes and highlighting the challenges in proving it generally.

Contribution

The paper proves the conjecture for several classes of graphs and discusses the difficulties in establishing a general proof.

Findings

Conjecture holds for bipartite, regular, and complete q-partite graphs.

No counter-examples found despite extensive searches.

Discussion of challenges in proving the conjecture universally.

Abstract

A well known upper bound for the spectral radius of a graph, due to Hong, is that $\mu_1^2 \le 2m - n + 1$. It is conjectured that for connected graphs $n - 1 \le s^+ \le 2m - n + 1$, where $s^+$ denotes the sum of the squares of the positive eigenvalues. The conjecture is proved for various classes of graphs, including bipartite, regular, complete $q$-partite, hyper-energetic, and barbell graphs. Various searches have found no counter-examples. The paper concludes with a brief discussion of the apparent difficulties of proving the conjecture in general.