Distance-based Topological Indices Of Graphs

Chemical graph theory is an interwined theory between quantum Chemistry and Graph theory.The distance-based topological indices(for example,Wiener index and Harary index,etc.)of molecular graphs reveal the relationships between chemical properties of molecular and graphic structure and consist of the main contents of chemical graph theory.This paper establishes the relationships among the Wiener and Harary indices and their variants and graphic parameters,such as diameter,the matching number,degrees,degree sequence,etc.Further,all graphs with the extremal values for these indices have been characterized.The main contents of this paper can be summarized as the following.In Chapter one,the backgrounds and some preliminaries on distance-based chem-ical indices are introduced and the main results are presented.In Chapter two,firstly,a sharp upper and lower bounds for the terminal Wiener index in terms of its order and diameter are presented and all extremal trees which attain these bounds are characterized.In addition,we investigate the properties of extremal trees which attain the maximum terminal Wiener index among all trees of order n with fixed maximum degree.Secondly,we present some formulae for the Wiener index of generalized Bethe trees,which correct the errors of[45]and a formula for the terminal Wiener indices of trees is obtained.With the formula,the terminal Wiener index of a general Bethe tree is presented,which corrects the errors of 45].Moreover,the trees with the minimum terminal Wiener index among all the trees of order n and with maximum degree ? are characterized in terms of graph transformation.In Chapter three,motivated by the study of local version of many graph invariants such as the Wiener index and the number of subtrees yielding interesting results on the"middle part" of a tree as well as extremal structures regarding the ratio of local func-tions at extremal vertices,we firstly consider the parallel questions when an additive parameter is treated as a local function and examine its behavior in terms of the "mid-dle part" of a tree and extremal values of the ratios and get some general results which can be generalized to Wiener index and mean hook-length of tree as a local version.Secondly,we consider the local version of the hyper-Wiener index(WW(G)),defined as wwG(v)=u?V(G)?(d2(u,v)+d(u,v))for a vertex in a graph G.For established results on the Wiener index(W(.)),we present analogous studies on WW(.)and study its behavior in terms of the "middle part" of a tree and extremal values of the ratios on wwT(w)/wwT(u)and wwT(w)/wwT(v).In addition to interesting observations,some conjectures and questions are also proposed.In Chapter four,three generalized topological indices,based on distance,degree,edge degree are proposed and have been investigated.With aid of Tutte-Berge formula,all extremal graphs with the maximum/minimum three indices respectively have been characterized.These results generalize/strength some known results.In Chapter five,we present some formulae for the first and second Zagreb indices of F-sums of two graphs in terms of the original graphs.Moreover,some exact expressions of the Zagreb indices of F-sums graphs and their applications are presented.