The Inertia Index Of Quasi Complete K-Partite Graphs

Algebraic graph theory is an important part of graph theory,and its research results are widely used in other fields.Adjacency spectral theory is an important research content of algebraic graph theory and is closely related to matrix theory.The rank of G,written as r(G),is defined to be the rank of A(G).The positive inertia index p(G),the negative inertia index n(G)and the nullityη(G)of G are respectively defined to be the number of positive eigenvalues,negative eigenvalues and zero eigenvalues of A(G).The positive and negative inertia index and nullity of the graph are collectively referred to as the inertia index of the graph,and have been widely used in the field of chemistry.In 1957,Collatz and Sinogowitz proposed to portray all singular graphs,which means all graphs with nullity greater than zero.In recent years,more and more graph theorists have begun to study the inertia index of graphs.In this article,we depict the inertia index of the quasi complete graph,the quasi complete bipartite graph,and the quasi complete tripartite graph,mainly including rank,positive and negative inertia index,nullity and signature.The whole text is divided into five chapters:The first chapter mainly introduces the research background and significance of graph theory and the inertia index of graph,introduces related concepts and symbolic explanations,and finally introduces the research issues of this thesis and the research progress of the inertia index of graphs at home and abroad in recent years.The second chapter depicts the inertia index of the quasi complete graph according to the connection between the points and points in the quasi complete graph by using the relevant lemma.The third chapter,we depict the inertia index of the quasi complete bipartite graph,using the method of removing twins points,simply transforming the quasi complete bipartite graph into a simpler graph,and depicting the inertia index of the quasi complete bipartite graph through its adjacency matrix.The fourth chapter mainly depicts the inertia index of the quasi complete tripartite graph,which is divided into four situations,and the method is the same as that of the third chapter.The fifth chapter summarizes the whole text and points out the innovation and dificiency of this thesis.Figure [13] Reference [78]...