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

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.

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