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

Abstract

a b s t r a c tGiven a graph G, the atom-bond connectivity (ABC ) index is defined to be ABCwhere 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.

Maximum atom-bond connectivity index with given graph parameters

“…In the study of topological indices in general, it is often of interest to consider the extremal values of a certain index among graphs under various constrains. Along this line, the extremal values of the ABC index have been extensively explored [2][3][4][5][6][9][10][11][12][13][14][15][16].…”

The structure of ABC-minimal trees with given number of leaves

“…A similar question has been investigated in [13,14,21,23,24], with intention to figure out which trees with the given number t of leaves (i.e. vertices of degree 1) have the minimum value of the ABC-index.…”

“…In this paper, a tree T is said to be t-minimal if T has t leaves and no other tree with the same number of leaves has smaller ABC-index. The problem of classifying t-minimal trees has been raised in [13] and [24] and was further explored in [14,21,23]. Magnant et al [24] claimed that t-minimal trees are balanced double stars whenever t ≥ 19.…”

“…The problem of classifying t-minimal trees has been raised in [13] and [24] and was further explored in [14,21,23]. Magnant et al [24] claimed that t-minimal trees are balanced double stars whenever t ≥ 19. This speculation was refuted by Goubko et al in [14],…”