The General Randi(?) Indices Of A Class Of Trees

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 zeroth-order general Randi(?) index of the graph G is defined as R_α⁰(G)=Σ_v∈Vd_v^α,where d_v is the degree of vertex v,αis an arbitrary real number. The general Randic index of the graph is defined as R_α(G)=Σ_uv∈E(G)(d_ud_v)^α.The zeroth-order general Randic index and general Randic index of the graph are the most important topological indices in chemical graph theory. They have a lot of applications in chemistry, and have been widely investigated as well.In the first chapter, we mainly introduce the zeroth-order general Randic index and general Randic index, provide the research advancement on them and give a brief overview to the main results of the thesis.In chapter 2, we shall investigate zeroth-order general Randi(?) index of a special types of tree with diameter not more than 4-the starlike tree S(c₁,c₂,…,c_d). Characterize completely the trees with the largest, the second largest, the third largest and the smallest zeroth-order general Randic index in S(c₁,c₂,…,c_d).In chapter 3, we shall investigate zeroth-order general Randic index of the thorn graph (?)^*{n,m) of (?)(n,m). Characterize completely extremal values of the zerothordergeneral Randic index of thorn graphs of trees, unicyclic graphs, bicyclic graphs. Extremal graphs are characterized as well.In chapter 4, we shall investigate the general Randi(?) index of the starlike tree S(c₁,c₂,…,c_d) whenα>0. Characterize completely the trees with the largest, the second largest, the third largest and the smallest general Randic index.In chapter 5, we shall investigate the general Randi(?) index of the starlike tree S(c₁,c₂,…,c_d) when -1≤α<0.Characterize completely the trees with the smallest, the second smallest, the third smallest and the largest general Randi(?) index.