Which Tree Has the Smallest ABC Index among Trees with K Leaves?
Discrete Applied Mathematics
Given a graph G, the atom–bond connectivity (ABC) index is defined to be ABC (G) = ∑u~v √ d(u)+d(v)-2/d(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.
Magnant, Colton, Pouria Salehi Nowbandegani, Ivan Gutman.
"Which Tree Has the Smallest ABC Index among Trees with K Leaves?."
Discrete Applied Mathematics, 194: 143-146.