Abstract
The investigation 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 in Congr Numer 60:187–197, 1987) or AutoGraphiX conjectures (Aouchiche et al. in: Liberti, Maculan (eds) Global optimization: from theory to implementation, Springer, New York, 2006). The reciprocal degree distance (RDD), the adjacent eccentric distance sum (AEDS), the average distance (AD) and the connective eccentricity index (CEI) are all distance-based graph invariants or topological indices, some of which found applications in Chemistry. In this paper, we investigate the relationship between RDD and other three graph invariants AEDS, CEI and AD. First, we prove that AEDS > RDD for any tree with at least three vertices. Then, we prove that RDD > CEI for all connected graphs with at least three vertices. Moreover, we prove that RDD > AD for all connected graphs with at least three vertices. As a consequence, we prove that AEDS > CEI and AEDS > AD for any tree with at least three vertices.
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