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Let $G$ be a graph. A minimal coloring of G is a coloring which
has the smallest possible sum among all proper colorings of $G$,
using natural numbers. The vertex-strength of $G$, denoted by
$s(G)$, is the minimum number of colors which is necessary to
obtain a minimal coloring. In this note we study these concepts,
and define a new concept called the edge-strength of $G$, denoted
by $s'(G)$. We pose some upper bounds for $s(G)$ using $\Delta
(G)$ and $col(G)$. Also, it is proved that $s'(G)$ lies between
$\Delta(G)$ and $\Delta (G)+1$, but it may not be equal to ${\cal
X}'(G)$. Based on our results on vertex-strength, we conjecture
that:
$$s(G)\leq \lceil\dfrac{{\cal X} (G)+\Delta(G)}{2}\rceil.$$
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