Some Problems On The Topological Indices And Laplacian Spectrum

Let G=(V(G),E(G)) be a graph with vertex set V(G) and edge set E(G).Two vertices of V(G) is said to be independent if they are not adjacent.Any subset of V(G) containing no two mutually adjacent vertices is called an independent set.The Merrifield-Simmons index is the sum of the number of independent set.Two edges of a graph G is said to be independent if they possess no common vertices. Any subset of E(G) containing no two mutually incident edges is called an independent edge set. Hosoya index is the sum of the number of the all independent edge sets. The authors [X.Z.Lv, Y.Y,A.M.Yu,J.J.Zhang, Ordering trees with given pendent vertices with respect to Merrifield-Simmons indices and Hosoya indices, J. Math. Chem.,2009] studied the order of trees with n vertices and k pendant vertices in the set Tn,k by means of Merriffield-Simmons index and Hosoya index. Based on their results, the trees with the first [n-k+1/2] largest Merrifield-Simmons index and the trees with the first [n-k+1/2] smallest Hosoya index are respectively determined.It is an important study field that we investigate the chemical topological indices for the unicyclic graphs in the mathematical chemistry. There are many results on this topic. Yu and Tian characterized m-matchings unicyclic graphs having the smallest and second-smallest Hosoya index and m-matching cyclic graphs having the smallest Hosoya index in [A.M.Yu,F.Tian, A kind of graphs with Minimal Hosoya indices and maximal merrifeld-Simmons indices, MATCH Commun. Math. Comput. Chem.,55(2006),103-118]. In this paper, we characterize unicyclic graphs of m-matchings from the third-smallest Hosoya index to fifth-smallest Hosoya index.Let A(G) be the adjacency matrix of G. Then the energy of G is defined to the sum of absolute values of its eigenvalues. It is quite interesting and useful to inves-tigate the energy of hydrocabons, since it is closely related to the total π-clection energy. In1977, Gutman first defined the relation "(?)" of bipartite graphs. Using this relation, we can efficiently solve many problems of graphs energy. However, some problems are still open. Recently, Huo solved several ones by the methods of analysis, algebra and combination and Coulson integral formula. The authors, char-acterized the graphs with minimal energy in the set Tn,k consisting of the trees with n vertices and k pendant vertices in [A.M. Yu, X.Z. Lv, Minimum energy ont trees with k pendent vertices, Linear Algebra and its Applications.,418(2006),625-633]. Using the similar method, the graphs with second minimal energy are determined in this paper. The authors [F.Li,B. Zhou, Minimal energy of unicyclic graphs of a given diameter, J. Math. Chem.,43(2008),476-484] characterized the graphs with the minimal energy and given diameter in the set of unicyclic graphs. In this thesis, the graphs with the second minimal energy and given diameter are determined.The Laplacian matrix of G is defined as L(G)=D(G)--A(G),Where D(G)=diag(d1,d2,…,dn) is the degree matrix of G.The Laplacian spectrum of G is a multiset consisting of its cigenvalues. In this paper, the spectral characterization of union of graphs with Laplacian spectral raidus at most4are completed solved.