Chromatic Zagreb indices for graphical embodiment of colour clusters

TL;DR

This paper explores the chromatic Zagreb indices for specific graph structures that embody colour clusters, focusing on their minimal edge configurations and chromatic properties.

Contribution

It introduces new graph models representing colour clusters and analyzes their chromatic Zagreb indices, highlighting their edge-minimality and chromatic characteristics.

Findings

Derived formulas for chromatic Zagreb indices of the proposed graphs.

Established the minimal edge count for graphs with given chromatic number.

Demonstrated the inverse relationship between edge-minimality and chromatic number.

Abstract

For a colour cluster $\mathbb{C} =(\mathcal{C}_1,\mathcal{C}_2, \mathcal{C}_3,\ldots,\mathcal{C}_\ell)$, 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 $G^{\mathbb{C}}_1$, $G^{\mathbb{C}}_2$ which represent graphical embodiments of the colour cluster such that the chromatic numbers $\chi(G^{\mathbb{C}}_1)=\chi(G^{\mathbb{C}}_2)=\ell$ and $\min\{\varepsilon(G^{\mathbb{C}}_1)\}=\min\{\varepsilon(G^{\mathbb{C}}_2)\} =\sum\limits_{i=1}^{\ell}r_i-1$. 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 $G^{\mathbb{C}}_1$, $G^{\mathbb{C}}_2$.