Coloring Sums of Extensions of Certain Graphs

TL;DR

This paper introduces and explores the concepts of hi'-chromatic sum and hi^+-chromatic sum for extended paths, cycles, and patterned structured graphs, expanding the understanding of graph coloring sums.

Contribution

It extends the definitions of hi'- and hi^+-chromatic sums to new classes of extended graphs beyond regular graphs.

Findings

Defined hi'- and hi^+-chromatic sums for extended paths and cycles.

Analyzed these sums for patterned structured graphs.

Provided new insights into graph coloring sums for extended graph classes.

Abstract

Recall that the minimum number of colors that allow a proper coloring of graph $G$ is called the chromatic number of $G$ and denoted by $\chi(G).$ In this paper the concepts of $\chi$'-chromatic sum and $\chi^+$-chromatic sum are introduced. The extended graph $G^x$ of a graph $G$ was recently introduced for certain regular graphs. We further the concepts of $\chi$'-chromatic sum and $\chi^+$-chromatic sum to extended paths and cycles. The paper concludes with \emph{patterned structured} graphs.