The Topological Indices Of Two Types Tricyclic Graphs

Let G=(V, E) be a simple connected graph with the vertex set V(G) and the edge set E(G). V(G)=nand E(G)=m denote the number of edges and vertices, respectively. A tricyclic graph is a connected graph with n vertices and n+2edges.Let S be a subset of the vertex set of G, if any two vertices in S are not adjacent then S is a independent vertex set of G. The Merrifield-Simmons index of the graph G, denoted by i(G), is the number of the independent vertex set of G. Let m(G, k) be the number of k-matchings of graph G. Then the Hosoya index of G is defined as The Wiener index W(G) of G is the sum of distances between all pairs of vertices in G, i.e where dG(u,v) is the distance between vertices u and v in G.Merrifield-Simmons index, Hosoya index and Wiener index are three typi-cal Topological indices. They have been widely investigated in mathematics and chemistry as well, and the Wiener index has been successfully used in theoretical chemistry for quantitative structure-property relations(QSPR) and quantitative structure-property relations(QSAR), and also in studying communication net-works.In this paper, we study the Merrifield-Simmons index, the Hosoya index and the Wiener index of some tricyclic graphs which contain three cycles. Firstly, we give some graph transformations, and then by these graph transformations, we characterize graphs with the largest, the second smallest Hosoya index, the small-est Merrifield-Simmons index and the smallest, the second smallest Wiener index, respectively. At the end of this paper, we get a part of results on the Merrifield-Simmons index, Hosoya index and Wiener index of some tricyclic graphs which contain four cycles.