| The Wiener index is one of the classical topological indices,which reflects the structure and properties of chemical molecules.It has a wide range of applications and has very important research significance.Since the Wiener index was proposed by the chemist Harold Wiener in 1947,it has become a hot topic of research.In recent decades,the research on the Wiener index has yielded fruitful results.In this paper,the mam research is on a generalized Wiener index,called as the square Wiener index,defined as the sum of the squares of the distances between all pairs of vertices of the graph,which is S(G)=Σu,vd2(u,v),where d(u,v)is the distance between the vertices u,v.The paper gives the research background,and research progress for Wiener index and square Wiener index.Then study the generalized Wiener index S(G),and give the relationship between S(G)and S(L(G))when G is a path,a circle or a graph with the minimum degree is at least 2.Similar to the Wiener index,the inertia indices(including the rank,nullity,positive and negative inertia index,signature of the adjacent matrix of a graph)are classical topological indices.With the relations with the spectrum of adjacent matrices,the inertia indices are also important algebraic parameters.According to quantum chemistry theory,the inertial index of the unsaturated hydrocarbon molecular map is closely related to the activeness of its molecular properties.The larger the eigenvalues of the molecular graph are.The smaller the amount of electricity,the less active the chemical properties.The larger the positive inertia index,the smaller the amount of electricity carried by the molecules.These findings make the research of the inertial indices become one hotspot among chemical graph theory and algebraic graph theory.The paper first gives the research background and progress of the inertia index.Then present an open conjecture about the signature:For any simple graph,-c3(G)≤s(G)≤c5(G).We confirm the conjecture for quasi-complete graphs and quasi-complete bipartite graph,furthermore,we completely determine the signature of pseudo-complete graphs.We also characterizes trees,unicyclic and bicyclic graphs with positive inertia index p(G)=2.Unicyclic graphs containing a triangle with rank of 6 are also determined explicitily.Figure [17] table [0] reference [56]... |
