Extremal Characterization Of Kirchhoff Index Of Graphs

The topological index of the molecule is a topological invariant of the molecular graph corresponding to the compound molecule.Molecular topological indices are of-ten used to describe the relationship between the molecular structure and properties of compounds.Kirchhoff index is one of the most important topological indexes.In 1993,Klein and Randic firstly proposed the concept of resistance distance in resistance net-works.By replacing each edge of graph G with unit resistor,a pure resistor network N graph was constructed.Based on the concept of resistance distance,Klein and Randid proposed the Kirchhoff index.The Kirchhoff index of graph G is defined as the sum of resistance distances between all vertices pairs in graph G.This paper focuses on on the issues of the Kirchhoff index of graphs and the structural properties of extremal graphs,as follows:The first part is introduction,which mainly introduces the research background,significance and research status of the Kirchhoff index of graphs,and the basic concepts and notations involved in this paper.The second part mainly studies the extremal graph of the Kirchhoff index of con-nected graphs given the number of cut vertex.First of all,we give the basic properties and related operations of the Kirchhoff index of the graph.Then the structural features of the extremal graph with the smallest Kirchhoff indicator in the n order connected graph and the given k(1 ≤ k ≤n/2)cut vertices are characterized.The third part mainly studies the upper bound of the Kirchhoff index of the uni-cyclic graph with exactly two maximal degree vertices and the corresponding extremal graph structure when the upper bound is reachable.The last part summarizes the main content of the paper.