On the maximal connective eccentricity index of bipartite graphs with some given parameters

The connective eccentricity index is a novel graph invariant with vast potential in structure activity/property relationships. This graph invariant displays high discriminating power with respect to both biological activity and physical properties. Given a simple connected graph G, the connective eccentricity index (CEI) of G is defined as ξee(G)=∑uv∈EG(1εG(u)+1εG(v)), where εG(⋅) denotes the eccentricity of the corresponding vertex. In this paper, we first determine the sharp upper bound on the CEI of graphs in the class of all n-vertex connected bipartite graphs with matching number q, the maximum CEI is realized only by the graph Kq,n−q. Second, we characterize the graph with the maximum CEI in the class of all the n-vertex connected bipartite graphs of given diameter. Finally, all the extremal graphs having the maximum CEI in the class of all the connected n-vertex bipartite graphs with a given connectivity s are identified as well.