On the Maximal Eccentric Distance Sums of Graphs.

If G is a simple connected graph with vertex V (G), then the eccentric distance sum of G, denoted by ξ^d (G), is defined as ∑_v∈V(G)ec_G(v)D_G(v), where ec_G(v) is the eccentricity of the vertex v and D_G(v) is the sum of all distances from the vertex v. Let n ≥ 8.We determine the n-vertex trees with, respectively, the maximum, second-maximum, third-maximum, and fourth-maximum eccentric distance sums. We also characterize the extremal unicyclic graphs on n vertices with respectively, the maximal, second maximal, and third maximal eccentric distance sums. [ABSTRACT FROM AUTHOR]