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The atom-bond connectivity (ABC) index of a graph G is defined as the sum over all pairs of adjacent vertices u;v, of the terms √ (d(u) + d(v) 2)=(d(u)d(v)), where d(v) denotes the degree of the vertex v of the graph G. Whereas the finding of the graphs with the greatest ABC-value is an easy task, the characterization of the graphs with smallest ABC-value, in spite of numerous attempts, is still an open problem. What only is known is that the connected graph with minimal ABC index must be a tree, and some structural features of such trees have been determined. Several conjectures on the structure of the minimal-ABC trees, were disproved by counterexamples. In this review we present the state of art of the search for minimal-ABC trees, and provide a complete bibliography on ABC index.

The ABC index conundrum