On the nullity of a connected graph in terms of order and maximum degree

Abstract

The nullity of a graph is the multiplicity of zero as an eigenvalue in its adjacency spectrum. Let G be a connected graph with n vertices, maximum degree Δ and nullity η. B. Cheng et al. (2020) proved that if G is not complete bipartite and Δ≥3, then η≤(Δ−2)nΔ−1. In this paper we prove that the said inequality for η becomes equality only if Δ=3, and identify all extremal graphs that attain the equality. As an immediate by-product, some connected graphs G with Δ=3 that satisfy η=n−12 are found. We work with 0-basic subgraphs and develop a new proof technique that is based on the concepts of dual vertex and pendant-dual vertex. Some open problems are also posed.