On the extremal eccentric connectivity index of graphs.

For a graph G = ( V , E ) , the eccentric connectivity index of G , denoted by ξ c ( G ), is defined as ξ c ( G ) = ∑ v ∈ V ɛ ( v ) d ( v ) , where ɛ( v ) and d ( v ) are the eccentricity and the degree of v in G , respectively. In this paper, we first establish the sharp lower bound for the eccentric connectivity index in terms of the order and the minimum degree of a connected G , and characterize some extremal graphs, which generalize some known results. Secondly, we characterize the extremal trees having the maximum or minimum eccentric connectivity index for trees of order n with given degree sequence. Finally, we give a sharp lower bound for the eccentric connectivity index in terms of the order and the radius of a unicyclic G , and characterize all extremal graphs. [ABSTRACT FROM AUTHOR]