A short proof of Zhou, Wong and Sun's conjecture

Abstract

Let G be a connected graph with order n, maximum degree Δ⩾2 and nullity η(G). Zhou, Wong and Sun [6] conjectured that η(G)⩽(Δ−2)n+2Δ−1 with equality if and only if G is a cycle Cn with n divisible by 4 or a complete bipartite graph Kn2,n2. Very recently, Cheng, Liu and Liu [2] confirmed this conjecture. In this paper, we give a much shorter proof for this conjecture. On the other hand, we show that η(G)⩽(Δ−2)nΔ−1 if Δ⩾3 and G∉{Kn2,n2,Kn+12,n−12}, which may deduce a main result obtained in [2].