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The ABC index conundrum

Gutman Ivan (Faculty of Science, University of Kragujevac, Kragujevac)
Furtula Boris (Faculty of Science, University of Kragujevac, Kragujevac)
Ahmadi Mohammad B. (Department of Mathematics, College of Sciences, Shiraz University, Shiraz, Iran)
Hosseini Seyyed A. (Department of Mathematics, College of Sciences, Shiraz University, Shiraz, Iran)
Nowbandegani Salehi Pouria (Department of Mathematics, College of Sciences, Shiraz University, Shiraz, Iran)
Zarrinderakht Maryam (Department of Mathematics, College of Sciences, Shiraz University, Shiraz, Iran)

The atom-bond connectivity (ABC) index of a graph G is defined as the sum over all pairs of adjacent vertices
,
, of the terms [d(u) + d(v)-2]/[d(u)d(v)] where d(v) denotes the degree of the vertex
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.

Keywords: Atom-bond connectivity index, ABC index, Trees, Extremal trees