Chromatic Zagreb indices for graphical embodiment of colour clusters

Abstract

For a colour cluster $\mathbb{C}=({\mathcal{C}}_1,{\mathcal{C}}_2,{\mathcal{C}}_3,\dots ,{\mathcal{C}}_{\ell })$, where ${\mathcal{C}}_i$ is a colour class such that $|{\mathcal{C}}_i|=r_i$, a positive integer, we investigate two types of simple connected graph structures $G_1^{\mathbb{C}}$, $G_2^{\mathbb{C}}$ which represent graphical embodiments of the colour cluster such that the chromatic numbers $\chi (G_1^{\mathbb{C}})=\chi (G_2^{\mathbb{C}})=\ell$ and $\min \{\epsilon (G_1^{\mathbb{C}})\}=\min \{\epsilon (G_2^{\mathbb{C}})\}=\sum _{i=1}^{\ell }{r}_i-1$. Therefore, the problem is the edge-minimality inverse to finding the chromatic number of a given simple connected graph. In this paper, we also discuss the chromatic Zagreb indices corresponding to $G_1^{\mathbb{C}}$, $G_2^{\mathbb{C}}$.

15 pages, 2 figures, communicated