Abstract
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.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.