On The Randi(?) Index Of Some Graphs

Being a booming discipline, graph theory has been developing rapidly in recent years because of its wide applications in various fields. The research of Randic indices has become one of the most important topic in graph theory. Randic index is an important chemical topological index which has a close relationship with many chemical properties. Afterwards, B.Bollobas and Erdos generalized it to general Randic index. The research on Randic index itself is a typical application of extremal graph theory. This thesis mainly studies sharp lower bounds of the Randic index of certain families of graphs.In Part 1, we introduce the developing background and some concepts of graph theory first. After that, we present the the definition and the background of Randic index, the main project now being studied and the achievement.In Part 2, we study the extreme value of the Randic index of (n,n+2)-graphs first. A graph is called a (n,n+2)-graph if it has n vertices and n+2 edges. We give sharp lower bounds on the Randic index of (n,n+2)-graphs. Next, we study the extreme value of the Randic index of a certain family of graphs in (n,n+r-1)-graphs. These graphs are those simple and connected ones obtained by adding a path of length k+1 to a cycle in a cactus with n-k vertices and r-1cycles, where r≥2, k≥0. We give sharp lower bounds on the Randic index of this class of graphs.At last, we conclude our research in this paper and propose some questions that are worthy of further in-depth consideration.