The Merrifield-Simmons Index And Hosoya Index For Cycle-linkage Graphs And Lexicographic Product Graphs

The Merrifield-Simmons index of a graph is defined as the total number of its independ-ent sets. The Hosoya index of a graph is defined as the total number of its matchings. Linkage graph is obtained by connecting several graphs in specific ways. Arithmetic graph is gained through structure operation of some graphs. There are two main ways to form the graph:the connection of graphs as well as structure operation of graphs. By connecting several loops in different ways we could get a plane graph called "cycle-linkage graph".In this paper, we study the counting and ordering problem of Merrifield-Simmons index and Hosoya index for cycle-linkage graphs and lexicographic product graphs.First and foremost, this essay forms four special graphs by four unique connecting ways: unilateral connection, two adjacent corresponding connections, coincides with single point, unilateral coincide according to the ring sequence of different orders. Then, this essay studies the counting of them for Merrifield-Simmons and Hosoya index under the different connection. The following rectangular Xn represents the order vector of the cycles, that is Xn =(m1,m2,…,mn). The rectangular Yn-2 represents the distance vector of the adjacent cycles, that is Yn-2=(k1,k2,…,kn-2).Secondly, this essay researches into the Merrifield-Simmons and Hosoya index for one of the subclasses of the ring cycles Γi(Xn,Yn-2)(i=1,2,3,4) and gain the Γi(1,1,…1) and Tj (2,2,…,2) as the pole graph of Tj(Yn-2), at the same time, it analyzes the arrangement of Tj(k, k,…,k), one of the subclasses of Tj(Yn-2) when k1=k2=…= kn-2. The concrete results are as follows:i) σ(Tj(1,1,…,1))<σ(Tj(3,3,…,3))<σ(Tj(5,5,…,5))<…< σ(Tj(6,6,…,6 ))<σ(Tj(4,4,…,4))<σ(Tj(2,2,…,2)).ii) μ(Tj(2,2,…,2))<μ(Tj(4,4,…,4))<μ(Tj(6,6,…,6))<. <μ(Tj(5,5,…,5))<μ(Tj(3,3,…,3))<μ(Tj(1,1,…,1)).Lastly, by researching into the counting and ordering of Merrifield-Simmons index for lexicographic product constructed road, ring, binary star, caterpillar tree and other special graphs with arbitrarily graph, this essay therefore comes at the counting expression and arrangement results of the Merrifield-Simmons index.