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 $\chi (G)\mathrm{.}$ 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.