Characterizing Graphs with Nullity N-4

The nullity of a graph G, denoted by η(G), is the multiplicity of the eigenvalue zero in the spectrum of G. A unified approach is presented for the characterization of graphs of order n with η(G) = n−4. All known results on trees, unicyclic graphs, bicyclic graphs, graphs with minimum degree 1, and r-partite graphs, for which η(G) = n−4 are shown to be corollaries of a theorem of Chang, Huang and Yeh that characterizes all graphs with nullity n − 4.