Certain Chromatic Sums of Some Cycle Related Graph Classes

TL;DR

This paper investigates various chromatic sums, including $ ext{chi}$-chromatic and $b$-chromatic sums, for standard cycle-related graphs, extending the understanding of graph coloring sums in these classes.

Contribution

It introduces new results on chromatic sums specifically for cycle-related graphs, expanding the existing literature on graph coloring sums.

Findings

Derived formulas for chromatic sums of cycle graphs

Extended results to related cycle-based graph classes

Provided bounds and exact values for specific chromatic sums

Abstract

Let $\mathcal{C} = \{c_1,c_2, c_3, \ldots,c_k\}$ be a certain type of proper $k$-colouring of a given graph $G$ and $\theta(c_i)$ denote the number of times a particular colour $c_i$ is assigned to the vertices of $G$. Then, the colouring sum of a given graph $G$ with respect to the colouring $\cC$, denoted by $\omega_{\cC}(G)$, is defined to be $\omega(\cC) = \sum\limits_{i=1}^{k}i\,\theta(c_i)$. The colouring sums such as $\chi$-chromatic sum, $\chi^+$-chromatic sum, $b$-chromatic sum, $b^+$-chromatic sum etc. are some of these types of colouring sums that have been studied recently. Motivated by these studies on certain chromatic sums of graphs, in this paper, we study certain chromatic sums for some standard cycle related graphs.