The Kirchhoff Index And Degree Kirchhoff Index Of Cacti

In this paper, by replacing each edge of G with a unit resistor, we consider G as an electrical network N. The resistance distance between v1 and V2, denoted by RG(vi, vj), is the effective resistance between vertices ui and vj. The Kirchhoff index of a graph is defined as the sum of resistance distance between all pairs of vertices in G. The degree Kirchhoff index of a graph is the sum of products between degree of all pairs of vertices and their resistance distances. A connected graph G is called a cactus if each block of G is either an edge or a cycle, both unicyclic graphs and tree are cacti.There are four chapters in this paper:In the first chapter, we mainly introduced some background and basic knowl-edge, as well as the research progress. The main research contents of this paper are also given in this chapter.In the second chapter, by introducing the five transformation operations and some lemmas,the maximum kirchhoff index of cacti is obtained, and the correspond-ing extremal graph is characterized as well.In the third chapter, we shall study the degree maximum kirchhoff index of cacti.In the fourth chapter, the main results of this paper are summarized, and on the basis of that, some further research directions are put forward.