| For a graph G,the independent sets counting index(abbreviation M-S index)and the matching counting index(abbreviation H index)are defined as the total number of its independent sets and the total number of its matchings,respectively.The M-S index and the H index are two prominent examples of topological indices which are of interest in structural chemistry.The study of these two topological indices mainly involves counting problems and ordering problems.The construction of graph structure is an important tool for studying these indices,there are two common methods:the linkage graph is obtained by connecting several graphs in specific ways,arithmetic graph is gained through structure operation of some graphs.In this paper,we study the counting and ordering problem of independent sets counting index and matching counting index for special copy-linkage graphs and some product graphs.(i)Inspired by the vertex connection methods of graph classes such as hexagonal systems,cycle-linkage graph,and computer interconnection network construction methods,four special copy-linkage graph Qmi(n,k),Pm,s,t r,j(n,k),Cm,s,t r,j(n,k),Tr(m1,m2,m3),where i=1,2,3,4,j=1,2,3,are defined using special connection ways of graph copy.(ii)We studied the independent sets and matching counting problem of special copy-linkage graph Qmi(n,k),i=1,2,3,4 and corona product and edge corona product graph,and the corresponding counting expression are given.Where,the independent sets and matching counting of corona product and edge corona product are an extension of some research results in the existing literature.(iii)For k＝1,2,…,r,we studied the ordering of special copy-linkage graphs Pm,s,t r,j(n,k),Cm,s,t r,j(n,k)with respect to independent sets and matching counting indices,the results show that:their ordering of the independent sets and the matching counting indices is exactly the opposite,the specific results are as follows:σ(Pm,s,t r,j(n,1))>σ(Pm,s,t r,j(n,3))>…>σ(Pm,s,t r,j(n,r-(r+1)2))>σ(Pm,s,t r,j(n,r-(r)2))>...>σ(Pm,s,t i,j(n,4))>σ(Pm,s,t r,j(n,2)μ(Pm,s,t r,j(n,1))<μ(Pm,s,t r,j(n,3))<...<μ(Pm,s,t r,j(n,r-(r+1)2))<μ(Pm,s,t r,j(n,r-(r)2))<...<μ(Pm,s,t r,j(n,4))<μ(Pm,s,t r,j(n,2))σ(Cm,s,t r,j(n,1))<σ(Cm,s,t r,j(n,3))<...<σ(Cm,s,t r,j(n,r-(r+1)2))<σ(Cm,s,t r,j(n,r-(r)2))<...<σ(Cm,s,t r,j(n,4))<σ(Cm,s,t r,j(n,2))μ(Cm,s,t r,j(n,1))>μ(Cm,s,t r,j(n,3))>...>μ(Cm,s,t r,j(n,r-(r+1)2))>μ(Cm,s,t r,j(n,r))>...>μ(Cm,s,t r,j(n,4))>μ(Cm,s,t r,j(n,2))where j=1,2,3,r=n-m-s-t+2.In addition,according to the definition of the matching energy and the related properties of the matching polynomials,the ordering of special copy-linkage graph Tr(m1,m2,m3)with respect to matching counting index is studied,and the corresponding sort results. |
