Research On The Connective Eccentricity Index Of Graphs With Some Given Parameters

Let G = (V(G),E(G)) be a simple connected graph with vertex set V(G)and edge set E(G). The connective eccentricity index (CEI) of G is defined as?ce(G) =?u?Vd(u)/?(u), where ?(u) is the eccentricity of v,while d(v) denotes the degree of v. This novel graph invariant was first proposed by Gupta, Singh and Madan in 2000. It has vast potential in structure activity/property relationships. This graph invariant displayed high discriminating power with regard to both biological activ-ity and physical properties. It attracts more and more researchers' attention. In this thesis, we study some extremal problems on the connective eccentricity index of graphs with some given parameters. All the corresponding extremal graphs are characterized.The concrete content is in the following:· In Chapter 1, we introduce the background.· In Chapter 2, we give some necessary definition, notations and terminologies and some necessary lemmas.· In Chapter 3, we determine the sharp upper bound on the CEI of n-vertex connected bipartite graphs with matching number.· In Chapter 4, we characterize the graphs with the maximum CEI among the class of all n-vertex connected bipartite graphs with given diameter and con-nectivity.· In Chapter 5, we characterize all the extremal graphs having the maximum CEI among all the n-vertex connected bipartite graphs with a given connectivity.· In Chapter 6, we characterize the graph with the maximum CEI among all the n-vertex connected graphs with given the minimum degree and connectivity.· In Chapter 7, we summarize the main results in this thesis and give some prospects for further research.