On the nullity of general graphs

The nullity of a graph G, denoted by eta(G), is the multiplicity of the eigenvalue zero in its spectrum of the adjacent matrix of the graph. It is known that if a graph G contains at least one edge, then eta(G) <= n - 2. Chen and Liu characterized general graphs with eta(G) is an element of {n - 2, n - 3}. Recently, the graphs with eta = n - 4, among some given classes of graphs, such as unicyclic graphs, bicyclic graphs, bipartite graphs and chordal graphs are determined. Li determined the extremal graphs with delta = 1 and eta(G) = n - 4 or n - 5. This paper mainly provides a necessary condition for graphs with nullity n - 5 and minimum degree delta, by which some known results ere obtained.