Comparative results and bounds for the eccentric-adjacency index

The eccentric-adjacency index (EAI) and the adjacent eccentric distance sum (AEDS) are two eccentricity-based graph invariants, both of which have a vast potential for predicting the physico-chemical and/or biological properties of molecules in QSAR/QSPR studies. More recently, Malik (2018) computed these two graph invariants for the join and corona products of graphs. In this paper, we present sharp upper bounds on EAI and investigate the relationship between EAI and AEDS. We first establish some sharp upper bounds on EAI for general connected graphs and quasi-trees. Then we investigate the relationship between AEDS and EAI, and we prove that AEDS>EAI for any tree with at least three vertices. Finally, we give two sufficient conditions for connected graphs satisfying the inequality AEDS>EAI and the inequality EAI>AEDS, respectively.