Bounds on the difference between the eccentric distance sum and the degree distance of graphs

Abstract

The eccentric distance sum and degree distance have been well-studied in the past several years. More recently, many authors have considered the relationships between several distance-based graph invariants. Hua et al. \cite{Hua1} investigated the relationship between the eccentric distance sum $\xi^d(G)$ and the degree distance $D'(G)$ of a graph $G$. In this paper, we give some further results on $\xi^d(G)-D'(G)$. Firstly, we determine upper and lower bounds on $\xi^d(G)-D'(G)$ among general connected graphs in terms of the number of cut edges, and characterize the corresponding extremal graphs. Meanwhile, we identify the extremal graphs of given girth $g$ having the minimum and maximum $\xi^d(G)-D'(G)$.  Secondly, we consider the extremal problems among bipartite graphs on $\xi^d(G)-D'(G)$ in terms of matching number. And then we characterize the extremal bipartite graphs with diameter $d$ having minimum $\xi^d(G)-D'(G)$.