Relationships between some distance–based topological indices

The Harary index (HI), the average distance (AD), the Wiener polarity index (WPI) and the connective eccentricity index (CEI) are distance–based graph invariants, some of which found applications in chemistry. We investigate the relationship between HI, AD, and CEI, and between WPI, AD, and CEI. First, we prove that HI > AD for any connected graph and that HI > CEI for trees, with only three exceptions. We compare WPI with CEI for trees, and give a classification of trees for which CEI ≥ WPI or CEI < WPI. Furthermore, we prove that for trees, WPI > AD, with only three exceptions.