Nullities of graphs with given order, matching number and cyclomatic number revisited

For a (simple) graph G, we denote by |V(G)|,|E(G)|,η(G) and m(G) respectively the order, the number of edges, the nullity, and the matching number of G. It was shown by Wang and Wong (2014) that for every graph G, |V(G)|−2m(G)−c(G)≤η(G)≤|V(G)|−2m(G)+2c(G), where c(G):=|E(G)|−|V(G)|+θ(G) is the cyclomatic number of G, θ(G) being the number of components of G. Graphs with the maximal nullity condition have been characterized by Song et al. (2015), and graphs with the minimal nullity condition have also been characterized independently by Rula et al. (2016) and Wang (2016). Earlier Guo et al. (2009) had also shown that for a unicyclic graph G, η(G)−|V(G)|+2m(G) can take only one of the values −1,0 or 2, and they characterized these three types of unicyclic graphs. In this paper, exploiting the concepts of canonical star associated with a rooted tree, the canonical unicyclic graph associated with a unicyclic graph and a crucial subgraph of a graph, we correct, complete and extend the work of previous authors on this topic. We give a nontrivial proof for the fact that the crucial subgraphs of a graph, as introduced by Wang (2016), are unique up to isomorphism. More complete lists of characterizations for the three types of unicyclic graphs, for nonsingular unicyclic graphs, and for graphs with the minimal or maximal nullity conditions are found. It is proved that if c,n are given positive integers with n≥6c+2, then for any integer k,−c≤k≤2c,k≠2c−1, there exists a connected graph G of order n that satisfies c(G)=c and η(G)−|V(G)|+2m(G)=k; however, if k is replaced by 2c−1 then there is no graph G of any order that satisfies the corresponding conditions.