Research On The Resistance Distance-Based Graph Parameter And The Graph Structure

Graph theory mainly studies the internal structure contained in graphs. Spec-tral theory is an important field of algebraic graph theory and combinatorial matrix theory. It mainly uses to character the structural properties of graphs through the spectral parameter of matrices associated to their graph representations, and studies the intrinsic relations between the spectral parameters and structure of graphs.In this thesis, according to the eigenvalue (resp. Laplacian eigenvalue, normal-ized Laplacian eigenvalue)theory of graphs, we obtain the inner link between the resistance distance-based graph parameter and graph structure. The concrete con-tent is in the following:. In Chapter 1, we introduce the background and significance of the research, including the development of a representative at home and abroad regarding this aspect. Based on this research background and profound discussion, by using deep-going analysis, it fully shows the main work’s necessity and innovation.. In Chapter 2, we give some necessary definition, graph structure and lemmas.. In Chapter 3, a formula for the normalized Laplacian characteristic polynomial of l(G) (resp. s(G),r(G) and q(G))in terms of the normalized Laplacian char-acteristic polynomial of G and number of vertices and edges of G is developed and used to give a formula for the degree-Kirchhoff index and the number of spanning trees of l(G) (resp. s(G), r(G) and q(G)), where l(G) is the line graph of G, s(G) is the subdivision graph of G, r(G) is the graph obtained from G by adding a new vertex corresponding to each edge of G and by joining each new vertex to the end vertices of the corresponding edge, and q(G) is the graph obtained from G by inserting a new vertex into every edge of G and by joining by edges those pairs of these new vertices which lie on adjacent edges of G.. In Chapter 4, we first show the decomposition theorem of normalized Laplacian polynomial of a graph, then based on it, explicit closed formula of the degree-Kirchhoff index and the number of spanning trees of Ln (a linear hexagonal chain with n hexagons) are derived, respectively.. In Chapter 5, first, the graph G in Bn+(the set of all connected bipartite bicyclic graph with n vertices) with the maximal (resp. second maximal) EE(G) and K(G) are established, respectively. And then we find that the first two bicyclic graphs in Bn+according to these two orderings are mainly coincident. Where are the Estrada index and Kirchhoff index of graph G, respectively.· In Chapter 6, we summarize the main results in this paper and give some prospects for further research in the future.