Solution to a conjecture on the maximum [formula omitted] index of graphs with given chromatic number.

Abstract The atom-bond connectivity (A B C) index of a graph G = (V , E) is defined as A B C (G) = ∑ u v ∈ E (G) d u + d v − 2 d u d v , where d u , d v are the degrees of the vertices u and v in G , respectively. In this paper, we prove that among all n -vertex graphs with given chromatic number χ ≥ 3 , the Turán graph T n , χ is the unique graph having the maximum A B C index, which completely confirms a conjecture posed in Zhang et al. (2016). [ABSTRACT FROM AUTHOR]