The Merrifield-Simmons Index And Independent Polynomial Of Trees

Let G = (V,E) be a simple and connected graph with the vertex set V(G) and the edge set E(G). A subset S C V is called a independent set of G if no two vertices of S are adjacent in G. The empty setφis also a independent set of G. Let S_k bethe number of independent sets of G with k elements, thenσ(G)=sum from k=0 to n S_k is calledthe Merrifield-Simmons index orσindex, where n is the number of vertices in G. In order to study the distribution of cardinal number with respect to independent sets,Gutman and Harary introduced the independent polynomialwhere S₀=1.In this thesis, we mainly discuss on the ordering of starlike trees with respect to their Merrifield-Simmons indices, and prove that the double star is uniquely determined by its independent polynomial among all connected graphs and its independentpolynomial is also unimodality and locate its mode.