Coloring Sums of Extensions of Certain Graphs

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 $χ(G).$ In this paper the concepts of $χ$'-chromatic sum and $χ^+$-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 $χ$'-chromatic sum and $χ^+$-chromatic sum to extended paths and cycles. The paper concludes with \emph{patterned structured} graphs.

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