The Extremal Kirchhoff Index Of Two Graphs

The resistance distance rijbetween vertices viand vjof a connected graph G is computed as the effective resistance between nodes viand vjin the corresponding network constructed from G by replacing each edge of G with a unit resistor. Analogue to the Wiener index, In 1993 Klein and Randic introduced a new index-Kirchhoff index. The Kirchhoff index Kf(G) is de?ned as:Kf(G) =∑i<j rG(vi, vj). So far, it has been studied extensively.In this paper,based on the previous study of the topological index of graphs, we further research the Kirchhoff index of fully loaded bicyclic graph and characterize the the extremal graphs with minimal and maximal Kirchhoff index among all fully loaded bicyclic graph. We also do some research on give the minimal Kirchhoff index among all cacti with n vertices and k pendant vertices, and characterize the corresponding cacti.This thesis consists of three chapters. In the ?rst chapter, we introduce some de?-nitions and notations about graphs with Kirchhoff index. Then we simply reviewed the study background and current studies in the ?eld of Kirchhoff index. In the end of this chapter, the main results of this thesis are listed.A fully loaded bicyclic graph is a bicyclic graph with the property that there is no vertex with degree less than 3 in its two cycles. In the second chapter, we study in the the Kirchhoff index of loaded bicyclic graph and characterize the extremal graphs with minimal and maximal Kirchhoff index among all fully loaded bicyclic graph.A connected graph G is a cactus if each block of G is either an edge or a cycle.In the three chapter, we do some research on ?nd the methods of computing the minimal Kirchhoff index among all cacti with n vertices and k pendant vertices, and characterize the corresponding cacti.