Extreme Values Of Some Topological Indices Related To Eigenvalue Of Graph

Let G =(V(G),E(G))be a simple undirected graph of order n with vertex-set VG and edge-set EG,A(G)be its adjacency matrix,D(G)?diag(d1,d2,…,dn)be the diagonal matrix of vertex degrees.Then Q(G)=D(G)+A(G)is the signless Laplacian matrix of G.Let ?1,?2,…,?n,and q1,q2,…,qn be the eigenvalues of A(G)and Q(G),respectively.Then the Estrada index and signless Laplacian Estrada index of G is de.fnned as EE(G)=? i=1n e?i and SLBE(G)??i=1n eqi,respectively.The resistance distance RG(u,v)between two vertices u and v of G is defined to be the effective resistance between the respective two corresponding points of an electrical network,constructed so as to correspond to G,,such that the resistance of any two ad-jacent points is unity.The sum of the resistance distances between all pairs of vertices in graph G is called the Kirchhoff index of G.On the basis of the concept of Kirchhof-f index,Chen and Zhang introduced the multiplicative degree Kirchhoff index,defined as S'(G)=?{u,v)(?)VG dG(u)dG(v)RG(u,v)and Gutman et al.introduced the additive degree Kirchhoff index defined as DR(G)=?{u,v}(?)VG(dG(u)+dG(v))RG(u,vDue to the wide applications of the topological indices of graphs in the fields of complex networks and chemistry,research on them has become one of important research directions in graph theory.This dissertation mainly studied the extreme-value problems of the Estrada index,the signless Laplacian Estrada index,the multiplicative degree Kirchhoff index and the additive degree Kirchhoff index for some classes of graphs with given graph parameters.In Chapters 1 and 2,some history and background as well as a few of basic concepts in graph theory were introduced,Then the background and progress of the research in this paper were explained.In Chapter 3,the Estrada index among the unicyclic graphs with fixed diameter was studied.The graphs with maximum Estrada index among these graphs were characterized.In Chapter 4,the signless Laplacian Estrada index for two classes of graphs was studied.The graphs with maximum signless Estrada index among these graphs were characterized.In Chapter 5,the additive degree Kirchhoff index and multiplicative degree Kirchhoff ndex of graphs were studied.The graphs with the second third largest additive degree Kirchhoff index and multiplicative degree Kirchhoff index among two classes of graphs were characterized,respectively.Finally,the work of this dissertation was summarized and some issues worthy of further discussion was brought forward.