Which tree has the smallest ABC index among trees with k leaves?

Abstract

© 2015 Elsevier B.V. Given a graph G, the atom-bond connectivity (ABC) index is defined to be ABC(G)=a<inf>u∼v</inf>d(u)+d(v)-2d(u)d(v) where u and v are vertices of G, d(u) denotes the degree of the vertex u, and u∼v indicates that u and v are adjacent. Although it is known that among trees of a given order n, the star has maximum ABC index, we show that if k 18, then the star of order k+1 has minimum ABC index among trees with k leaves. If k≥19, then the balanced double star of order k+2 has the smallest ABC index.