Computer search for large trees with minimal ABC index

The atom-bond connectivity (ABC) index of a graph G = (V, E) is defined as ABC(G)=∑vivj∈E(di+dj−2)/(didj), where V = {v0,v1,⋅⋅⋅, vn − 1} and di denotes the degree of vertex vi of G. This molecular structure descriptor found interesting applications in chemistry, and has become one of the most actively studied vertex-degree-based graph invariants. However, the problem of characterizing n-vertex tree(s) with minimal ABC index remains open and was coined as the “ABC index conundrum”. In attempts to guess the general structure of such trees, several computer search algorithms were developed and tested up to n = 800. However, for large n, all current search programs seem too powerless. For example, the fastest one up to date reported recently in [30] costs 2.2 h for n = 800 on a single PC with two CPU cores. In this paper, we significantly refine the known features of the degree sequence of a tree with minimal ABC index. With the refined features a search program was implemented with OpenMP. Our program was tested on a single PC with 4 CPU cores, and identified all n-vertex tree(s) with minimal ABC index up to n = 1100 within 207.1 h. Some observations are made based on the search results, which indicate some possible directions in further investigation of the problem of characterizing n-vertex tree(s) with minimal ABC index.