  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.
