On the Relation Between Degree Distance and Eccentric Connectivity Index

The relation between the degree distance DD(C) and the eccentric connectivity index xi(c)(G) of a graph G is studied. Sharp lower and upper bounds on DD(G) involving xi(c)(C) are determined. A Nordhaus-Gaddum type result on DD(G) in terms of xi(c)(G) and the first Zagreb index is proposed for connected graphs G with connected complements. A sharp upper bound on DD(T) involving xi(c)(T) is given for trees of given order. It is proved that the difference DD(T) - xi(c)(T) on trees with the same order is minimized on caterpillars. Effect on the degree distance versus the eccentric connectivity index of a tree under edge contractions is also investigated.