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

A well known upper bound for the spectral radius of a graph, due to Hong, is that μ12≤2m−n+1 if δ≥1δ≥1. It is conjectured that for connected graphs n−1≤s+≤2m−n+1n−1≤s+≤2m−n+1, where s+s+ denotes the sum of the squares of the positive eigenvalues. The conjecture is proved for various classes of graphs, including bipartite, regular, complete qq-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.