An upper bound of the nullity of a graph in terms of order and maximum degree

Let G be a finite undirected graph without loops and multiple edges. By η,Δ and n we respectively denote the nullity, the maximum vertex degree and the order of G. In [10], it was proved that η≤n−2⌈n−1Δ⌉ when G is a tree. This result was generalized to a bipartite graph by [25]. For a reduced bipartite graph G, the above inequality was improved to n−2−2ln2Δ by [27]. However, the problem of bounding the nullity of an arbitrary graph G in terms of n and Δ is left open for more than ten years. In this article, we aim to solve such a left problem. We prove that η≤Δ−1Δn for an arbitrary graph G with order n and maximum degree Δ, and the equality holds if and only if G is the disjoint union of some copies of KΔ,Δ.