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The eccentric distance sum (EDS) and degree distance (DD) are two distance-based graph invariants which have been well-studied in recent years. The study on relationships between various graph invariants has received much attention over the past few decades, and some of these research are associated with Graffiti conjectures (Fajtlowicz and Waller, 1987) or AutoGraphiX conjectures (Aouchiche et al., 2006). More recently, several groups of authors have investigated the relationships between several distance-based graph invariants along this line, see e.g., Klavžar and Nadjafi-Arani (2014), Hua et al. (2015), and Zhang and Li (0000), and so on. In this paper, we investigate the relationship between the eccentric distance sum and degree distance. First, we establish several sufficient conditions for a connected graph to have a larger/smaller EDS than DD, respectively. Second, we investigate extremal problems on the difference between EDS and DD for general connected graphs, trees, and self-centered graphs, respectively. More specifically, we present sharp upper and lower bounds on the difference between EDS and DD among all connected graphs, trees and self-centered graphs, respectively. In addition, we characterize all extremal graphs attaining those upper or lower bounds. |
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