Relation between the nullity of a graph and its matching number

Let G be a connected graph of order n(G) and size e(G). The nullity of G, denoted by η(G), is the multiplicity of eigenvalue zero of the adjacency matrix of G. In 2014, Wang and Wong (2014) bounded η(G) as n(G)−2m(G)−c(G)≤η(G)≤n(G)−2m(G)+2c(G), where m(G) is the matching number of G and c(G), defined by c(G)=e(G)−n(G)+1, is the dimension of cycle space of G. We find that the upper and the lower bounds for η(G) both fail to accurate if the edges in G are dense, namely if c(G) is large enough. In this paper, we aim to improve the above bounds. It is proved that n(G)−2m(G)−σ(G)≤η(G)≤n(G)−2m(G)+2ω(G), where σ(G) is the largest number of disjoint odd cycles in G and ω(G) is the number of even cycles in G. The cycle-disjoint connected graphs with nullity n(G)−2m(G)−σ(G) are characterized.