Abstract
Let G be a graph containing no component isomorphic to the path graph of order 2. Denote by d(w) the degree of a vertex w in G. The augmented Zagreb index (AZI) of G is the sum of the quantities (d(u)d(v)/(d(u) +d(v) - 2))(3) over all edges uv of G. Denote by g(n, chi) the class of all connected graphs of a fixed order n and with a fixed chromatic number x, where n >= 5 and 3 <= chi <= n - 1. The problem of finding graph(s) attaining the maximum AZI in the class g(n, chi) was addressed recently in [F. Li, Q. Ye, H. Broersma, R. Ye, MATCH Commun. Math. Comput. Chem. 85 (2021) 257-274] for the case when n is a multiple of chi. The present paper gives the complete solution to the aforementioned problem.