Research On Merrified-Simmons Index Of Four Special Graphs

In graph theory,the sum of the independent set of graphs is called the Merrified-Simmons index(abbreviated as MS index)of graph G,and MS indicators are very important for predicting,synthesizing new compounds and new drugs.In this paper,we mainly study the MS indexes of four special graphs,including Complete K-bipartite Graphs,Star-like Tree,Special Unicyclic Graphs and special Caterpillars Graphs.Special Unicyclic Graphs is a common circle of chemical molecules common structure,and star-like tree and caterpillar tree can often saw in network structure.For the complete K-bipartite graphs,we obtain the expression of MS index of K-part graphs:2m1+2m2+…mk + 1-k,and prove the result is correct.For the unicyclic graph,because of the universality of its structure,it is difficult for us to obtain the expression of the MS index of the general unicyclic graph.So we defines two special unicyclic graphs in expansion,using the independent polynomial and the recurrence of the Fibonacci sequence Relationship,the more special two kinds of unicyclic graphs,get the upper and lower bounds of special unicyclic graphs finally.We found that if we get the largest graph of NI(G),then G must be Gn3Sn-3,and the smallest graph of NI(G)must be the result of Gn(3,3,3,…,3 + p).For the star-like tree,we mainly compare two different branches of the star-like tree,and sort the star-like trees with the same branches according to the MS index,and the pole diagram of the star tree can be described if we use the facts repeatedly.For the caterpillar graphs,because of the complexity of its structure,it is difficult for us to derive the MS indicator of the general caterpillar tree.In this paper,we construct a special caterpillar tree.By analyzing its structural features and using the method of combinatorial reasoning,the specific expression of MS index was calculated.The counting expression of MS marker was studied and the results were validated.