The nullity of a graph with fractional matching number

Abstract

Let G be a simple graph with n(G) vertices and e(G) edges. Denoted by η(G) and m⁎(G) the nullity and the fractional matching number of G, respectively. The dimension of cycle space of G is defined as c(G)=e(G)−n(G)+ω(G), where ω(G) denotes the number of connected components of G.In this paper, we prove that n(G)−2m⁎(G)≤η(G)≤n(G)−2m⁎(G)+2c(G) for a graph G, which improves the main results of Wang and Wong (2014) [19] and Ma and Fang (2019) [11], respectively. Furthermore, all graphs with nullity η(G)=n−2m⁎(G)+2c(G) are determined. We also prove that there is no graph with nullity η(G)=n−2m⁎(G)+2c(G)−1; and for fixed c(G), infinitely many connected graphs with nullity n−2m⁎(G)+2c(G)−k(0≤k≤2c(G),k≠1) are also constructed. As an application of the above results, we also prove that if G is nonsingular, then G has a fractional perfect matching.