On the extremal eccentric connectivity index of graphs

For a graph G=(V,E), the eccentric connectivity index of G, denoted by ξc(G), is defined as ξc(G)=∑v∈Vɛ(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.