On the maximal eccentric connectivity indices of graphs

For a connected simple graph G, the eccentricity ec(v) of a vertex v in G is the distance from v to a vertex farthest from v, and d(v) denotes the degree of a vertex v. The eccentric connectivity index of G, denoted by ξ ^c (G), is defined as Σ _v∈V(G) d(v)ec(v). In this paper, we will determine the graphs with maximal eccentric connectivity index among the connected graphs with n vertices and m edges(n ≤ m ≤ n+4), and propose a conjecture on the graphs with maximal eccentric connectivity index among the connected graphs with n vertices and m edges (m ≥ n +5).