Two Types Of Distance-based Topological Indices Of Graphs

The graph invariants are some special mappings from the graph to the set of real numbers,which remain their values unchanged in the isomorphic graphs.Some graph invariants,which are based on the distances between the vertices in graph,have been frequently used in the fields of biology,chemistry and physics.And as the topological indicators,the mathematical properties of these graph invariants have also received extensive attention from some scholars.There are many types of graph invariants.In this paper,we only focus on two of them:the eccentric distance sum and the degree Kirchhoff index.In 2002,in order to predict the chemical and physical properties of a compound better,Gupta,Singh and Madan created the eccentric distance sum,which was based on the distances between the vertices.For a simple connected graph G,the eccentric distance sum of G is denoted as?d(G)=?u,v?V(G)(?G(u)+?G(v))dG(u,v),in which the ?G(·)is the eccentricity of the corresponding vertex and the dG(u,v)is the distance between the vertices u and v.In 2007,when Chen and Zhang were focused on the study of the resistance distance in graph,they created the degree Kirchhoff index.For a simple connected graph G,the degree Kirchhoff index of G is denoted as S'(G)=?u,v?V(G)dG(u)dG(v)RG(u,v),in which the dG(·)is the degree of the corresponding vertex and the RG(u,v)is the resistance distance between the vertices u and v.In this paper,we mainly study the extreme values of these two graph invariants in some kinds of special graphs.For a simple connected graph G,we suppose that n:=| V(G)| and m:=| E(G)|.Ifm=n+1,then the G is called the bicyclic graph.If m=n+2,then the G is called the tricyclic graph.And if any two cycles of G have at most one common vertex,the graph G is called the cactus.These three types of graphs with special structures are ubiquitous in graph theory.Therefore,it is important to study the invariants of these graphs.The mathematical properties of the eccentric distance sum and the degree Kirchhoff index in special graphs have been studied deeply,and a large number of relevant conclusions have been obtained.In this paper,by using several edge-grafting transformations,we finally characterize the extremal bicyclic graphs,the extremal tricyclic graphs and the extremal cacti with the minimum eccentric distance sum.And on the other hand,we also completely characterize the bicyclic graphs having the maximum and the second-maximum degree Kirchhoff index.