The Kirchhoff Index Of Graphs

The resistance distance rij between vertices vi and Vj of a connected graph G is computed as the effective resistance between nodes vi and Vj in the corresponding network N, which is constructed from G by replacing each edge of G with a unit resistor. Klein and Randic defined the Kirchhoff index Kf(G) of G as the sum of resistance distances between all pairs of vertices in G. This is a very important topological index in Quantum Chemistry, which has been extensively studied.In this work, we determined the extremal problems of Kirchhoff index among all n-vertex graphs with given number of cut vertices and quasi-tree graphs, respectively. The first chapter we introduced the basic concepts and notations of graph theory involved in this paper. Then we pointed out the research backgrounds and surveyed the research developments of Kirchhoff index at home and abroad. In the second chapter, let gn,k be the class of connected graphs of order n and with k cut vertices. We attained the minimal Kirchhoff index of graphs from gn,k with 0≤k<n/2 and characterized the corresponding extremal graphs. In the third chapter, let 2T’(n) be the set of non-trivial quasi-tree graphs of order n. We obtained the maximal Kirchhoff index among all graphs from 2T’(n). Moreover, a lower bound on Kirchhoff index was proposed for any graph from 2T’(n). In the last chapter we listed some related open problems to this topic.