On a relation between the atom-bond connectivity and the first geometric–arithmetic indices

The atom-bond connectivity index (ABCABC) and the first geometric–arithmetic index (GAGA) are two well-known molecular descriptors, which are found to be useful tools in QSPR/QSAR investigations. In this work, we obtain a relation between these two indices for simple connected graphs on n≥3n≥3 vertices with minimum degree at least ss and maximum degree at most tt, where 1≤s≤t≤n−11≤s≤t≤n−1 and t≥2t≥2. Using this relation, we prove that if t≤4s2−3s+1t≤4s2−3s+1, then the ABCABC index is always less than the GAGA index and this bound is best possible for s≥2s≥2.