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The atom-bond connectivity (ABC) index is one of the recently most investigated degree-based molecular structure descriptors, that have applications in chemistry. For a graph G, the ABC index is defined as uv∈E(G), where d(u) is the degree of vertex u in G and E(G) is the set of edges of G. Despite many attempts in the last few years, it is still an open problem to characterize trees with minimal ABC index. In this paper, we present an efficient approach of computing trees with minimal ABC index, by considering the degree sequences of trees and some known properties of trees with minimal ABC index. The obtained results disprove some existing conjectures and suggest new ones to be set.

The atom-bond connectivity (ABC) index is a degree-based molecular descriptor, that found chemical applications. It is well known that among all connected graphs, the graphs with minimal ABC index are trees. A complete characterization of trees with minimal ABC index is still an open problem. In this paper, we present new structural properties of trees with minimal ABC index. Our main results reveal that trees with minimal ABC index do not contain so-called B k -branches, with k ≥ 5, and that they do not have more than four B 4 -branches.

Albertson [4] has defined the irregularity of a simple undirected graph G as irr(G) = uv∈E(G) |d G (u) − d G (v)| , where d G (u) denotes the degree of a vertex u ∈ V (G). Recently, in [1] a new measure of irregularity of a graph, so-called the total irregularity, was defined as irr tHere, we compare the irregularity and the total irregularity of graphs. For a connected graph G with n vertices, we show that irr t (G) ≤ n 2 irr(G)/4. Moreover, if G is a tree, then irr t (G) ≤ (n − 2)irr(G).