Research On One Distance-based Graph Parameter And The Characterization Of The Extremal Graph

In graph theory one mainly focuses on the internal structure contained in graphs.The relation between the distance-based parameters and the structures of graphs is an important rescarch field of modern graph theory.Let G = (VG,EG) be a simple connected graph. The reciprocal degree distance of G is defined as(?)where dG(·) is the vertex degree of the corresponding vertex,dG(u,v)is the distance between u and v in G and DG(u) ? ?v?VG\{u|1/dG(u,v) is the sum of reciprocal distance from the vertex u. This graph invariant was first proposed independently by Yaser Alizadeh, Ali Iranmanesh and Tomislav Doslic, and Shenggui Zhang, Hongbo Hua in 2013. After that more and more researchers studied this graph invariant. This graph invariant displayed high discriminating power with respect to biological activity and physical properties and some chemical molecular structures. It was used to investigate various physical properties and date sets of analogues. Hence, it is meaningful to study the reciprocal degree distance of graphs.The concrete content is in the following:· In Chapter 1, we introduce the background and research development on the reciprocal degree distance of a graph.· In Chapter 2, we give some necessary definitions, notations, terminologies and some necessary lemmas.· In Chapter 3, we determine the graphs among yn,dk that have the maximum R-value and characterize the structure of the extremal graphs.· In Chapter 4, we determine sharp upper bound for R(G) of graphs among yn,kl??0.Furthermore,the sharp upper bound for R(G) of graphs among yn,k?0 is also determined. All the extremal graphs are identified.· In Chapter 5, we determine the sharp lower bound on the R-value of n-vertex graphs with connectivity k and independence number ?.· In Chapter 6, we summarize the our main results in this paper and give some prospects for further study.