Chromatic number and some multiplicative vertex-degree-based indices of graphs

For a (molecular) graph, the rst and second Zagreb indices (M₁ and M₂) are two well-known topological indices in chemical graph theory introduced in 1972 by Gutman and Trinajstić. Multiplicative versions of Zagreb indices, such as Narumi-Katayama index, multiplicative Zagreb index and multiplicative sum Zagreb index, have been much studied in the past. Let G(n, k) be the set of connected graphs of order n and with chromatic number k. In this paper we show that, in G(n, k), Turan graph T_n(k) has the maximal Narumi-Katayama index, the maximal multiplicative Zagreb index and the maximal multiplicative sum Zagreb index. And the extremal graphs from G(n, k) with k = 2 or 3 are determined with minimal values of these above indices.