Chromatic Zagreb and irregularity polynomials of graphs

Graph coloring is an assignment of colors, labels or weights to elements of a graph subject to certain constraints. Coloring the vertices of a graph in such a way that adjacent vertices are having different colors is called proper vertex coloring. A proper vertex coloring using minimum parameters of colors is studied extensively in recent literature. In this paper, we define new coloring related polynomials, called chromatic Zagreb polynomials and chromatic irregularity polynomials, in terms of minimal parameter coloring and structural characteristics of graphs such as distances and degrees of vertices.