Ounds On The Zeroth-order General Randic Index For Two Classes Of Graphs

Let G=(V,E) be a simple connected graph, where V(G), E(G) denote the set of vertices and edges of G, respectively. The zeroth-order general Randic index of G is defined as: where dG(v) denotes the degree of the vertex υ of G and a is a real number. The zeroth-order general Randic index of the graph is one of the most important topo-logical indices in chemical graph theory. And it has excellent correlations with the physical and chemical indices of molecular compounds. Therefore, The zeroth-order general Randic index has intensive applications in many fields.In this paper, we investigate the bounds on the zeroth-order general Randic index for bicyclic graphs (?)n,dand tricyclic graphs (?)2m by using the graph sequence and graph transformations,respectively. Let (?)n,dand (?)2m be the set of simple connected bicyclic graphs with n vertices, diameter d and the set of conjugated tricyclic graphs with m vertices,2m edges, respectively.In Chapter one, we introduce the brief background of the zeroth-order general Randic index. Then we introduce some basic concepts and the results which are obtained in this thesis.In Chapter two, we give some lemmas for obtaining the important results. Then the maximal and minimal zeroth-order general Randic index of simple connected bicyclic graphs in (?)n,d are entirely characterized, and the graph with extremal values of the zeroth-order general Randic index is characterized.In Chapter three, we discuss the conditions of the conjugated tricyclic graphs (?)2m with three,four, six and seven cycles, respectively. Then we give the conjugated tricyclic graph with the maximal zeroth-order general Randic index for α>2.In chapter four, some problems to be studied further are discussed.