On The Nullity Of Signed Graphs

Let A(G) be the adjacency matrix of a graph G. The nullity of G is the multiplicity of the zero eigenvalue in the spectrum of A(G). If the nullity of a graph is no less than1, then we call G a nonsigular graph.The nullity has great utility. For example, it could be used to investigate the stability of the alternate hydrocarbons. The nullity of graphs has a lot of applications in many subjects, so it has attracted considerable attention. In fact, this issue has become one of hotspots of the spectrum theorem.A signed graph is a graph with a sign defined on each of its edges. Up till now, many themes of graph theory has extended to signed graph. We characterize the signed graphs of order n with great nullity, and further we define three graph transformations that preserve the nullity. Moreover, we determine the signed graphs that have identified structure with great nullity.Here we introduce the structure of this dissertation. In Chapter1, some def-initions, concepts and necessary conclusions are provided. First, the background and meaning of graph theory, spectral graph theory and the nullity of signed graph are pointed out. Then, some common basic concepts and necessary conclusions are presented in detail. At the last of this chapter, a brief introduction of the problem and the main conclusions of this dissertation will be provided. In Chapter2, we investigate the nullity of bicyclic signed graphs. First, the nullity of signed graphs is studied. The signed graphs of order n with nullity n-2or n-3are character-ized. Then, the unbalanced bicyclic signed graphs of order n with nullity n-3or n-4, and the signed graphs (including the simple bicyclic graphs) of order n with nullity n-5are characterized. In Chapter3, the nullity of tricyclic signed graphs are characterized. Some conclusions of bicyclic signed graph and a relation between bicyclic signed graph and tricyclic signed graph are introduced. Then the tricyclic signed graphs of order n, including a fixed signed subgraph, with nullity n-4or n-5are determined.