Proof of a conjecture on the nullity of a graph

Let G be a finite undirected graph without loops and multiple edges. The nullity of G, written as eta(G), is defined to be the multiplicity of 0 as an eigenvalue of its adjacency matrix. The left problem of establishing an upper bound for an arbitrary graph in terms of order and maximum degree was recently solved by Zhou et al. Zhou et al proved that eta(G)<=Delta-1 Delta n for an arbitrary graph G without isolated vertices and with order n, with maximum degree Delta >= 1, the equality holds if and only if G is the disjoint union of some copies of K Delta,Delta, and they posed a conjecture: If G is assumed to be connected, the upper bound of eta(G) can be improved to (Delta-2)n+2 Delta-1, and the upper bound is attained if and only if G is a cycle Cn with n divisible by 4 or a complete bipartite graph with equal size of chromatic sets. The goal of the present paper is to give a proof confirming the conjecture.