Chromatic Zagreb indices for graphical embodiment of colour clusters

<p>For a colour cluster <span class="math"><em>C</em> = (C<sub>1</sub>, C<sub>2</sub>, C<sub>3</sub>, …, C<sub>ℓ</sub>)</span>, where <span class="math">C<sub><em>i</em></sub></span> is a colour class such that <span class="math">∣C<sub><em>i</em></sub>∣ = <em>r</em><sub><em>i</em></sub></span>, a positive integer, we investigate two types of simple connected graph structures <span class="math"><em>G</em><sub>1</sub><sup><em>C</em></sup></span>, <span class="math"><em>G</em><sub>2</sub><sup><em>C</em></sup></span> which represent graphical embodiments of the colour cluster such that the chromatic numbers <span class="math"><em>χ</em>(<em>G</em><sub>1</sub><sup><em>C</em></sup>) = <em>χ</em>(<em>G</em><sub>2</sub><sup><em>C</em></sup>) = ℓ</span> and <span class="math">$\min\{\varepsilon(G^{C}_1)\}=\min\{\varepsilon(G^{C}_2)\} =\sum\limits_{i=1}^{\ell}r_i-1$</span>, and <span class="math"><em>ɛ</em>(<em>G</em>)</span> is the size of a graph <span class="math"><em>G</em></span>. In this paper, we also discuss the chromatic Zagreb indices corresponding to <span class="math"><em>G</em><sub>1</sub><sup><em>C</em></sup></span>, <span class="math"><em>G</em><sub>2</sub><sup><em>C</em></sup></span>.</p>