Ordering Unicyclic Graphs With Respect To Hosoya Index And Merrifield-Simmons Index

Let G = (V, E) be a simple and connected graph with the vertex set V(G) and the edge set E(G), |V(G)| = n, |E(G)| = m be the number of vertex and edge of G, respectively. The unicyclic graph is a connected graph which the vertex number is equal to the edge number.Let m(G, k) be the k-matching of graph G. Then the Hosoya index of G isdefined as z(G) = (?)m(G, k).Let S be a subset of the vertex set of G. If any two verties in S are not adjacent, then S is an independent vertex set of G. Let i(G) denote the Merrifield-Simmons index of the graph G. Then i(G) is the number of the independent vertex set of G.Hosoya index and Merrifield-Simmons are the most widely used topological indices of chemical graph theory. They have a lot of applications in chemistry, and have been widely investigated in mathematics as well.We shall investigate the Hosoya index and Merrifield-Simmons index of unicyclic graphs by some graph transformations, and get the first eighth smallest unicyclic graphs, and the second largest unicyclic graphs with respect to the Hosoya index, the first seventh largest unicyclic graphs with respect to Merrifield-Simmons index in this paper.