A characterization of graphs G with nullity |V(G)|−2m(G)+2c(G)|V(G)|−2m(G)+2c(G)

By |V(G)||V(G)|, |E(G)||E(G)|, η(G)η(G), and m(G)m(G) we denote respectively the order, the number of edges, the nullity, and the matching number of a (simple) graph G. Recently Wang and Wong have proved that for every graph G  , η(G)≤|V(G)|−2m(G)+2c(G)η(G)≤|V(G)|−2m(G)+2c(G), where c(G)=|E(G)|−|V(G)|+θ(G)c(G)=|E(G)|−|V(G)|+θ(G), θ(G)θ(G) being the number of connected components of G. In this paper graphs G   that satisfy the equality η(G)=|V(G)|−2m(G)+2c(G)η(G)=|V(G)|−2m(G)+2c(G) are characterized.