On chromatic Zagreb indices of certain graphs

Let [Formula: see text] be a finite and simple undirected connected graph of order [Formula: see text] and let [Formula: see text] be a proper vertex coloring of [Formula: see text]. Denote [Formula: see text] simply, [Formula: see text]. In this paper, we introduce a variation of the well-known Zagreb indices by utilizing the parameter [Formula: see text] instead of the invariant [Formula: see text] for all vertices of [Formula: see text]. The new indices are called chromatic Zagreb indices. We study these new indices for certain classes of graphs and introduce the notion of chromatically stable graphs.