Some extremal results on the connective eccentricity index of graphs

The connective eccentricity index (CEI) of a graph G is defined as ξce(G)=∑vi∈V(G)d(vi)ε(vi) where ε(vi) and d(vi) are the eccentricity and the degree of vertex vi, respectively, in G. In this paper we obtain some lower and upper bounds on the connective eccentricity index for all trees of order n and with matching number β and characterize the corresponding extremal trees. And the maximal graphs of order n and with matching number β and n edges have been determined which maximize the connective eccentricity index. Also the extremal graphs with maximal connective eccentricity index are completely characterized among all connected graphs of order n and with matching number β. Moreover we establish some relations between connective eccentricity index and eccentric connectivity index, as another eccentricity-based invariant, of graphs.