  We find a set of necessary and sufficient conditions under which the weight
$w:E\to\mathbb R^+$ on the graph $G=(V,E)$ can be extended to a pseudometric
$d:V\times V\to\mathbb R^+$. If these conditions hold and $G$ is a connected
graph, then the set $\mathfrak M_w$ of all such extensions is nonvoid and the
shortest-path pseudometric $d_w$ is the greatest element of $\mathfrak M_w$
with respect to the partial ordering $d_1 \leqslant d_2$ if and only if
$d_1(u,v) \leqslant d_2(u,v)$ for all $u,v\in V$. It is shown that every
nonvoid poset $(\mathfrak M_w,\leqslant)$ contains the least element
$\rho_{0,w}$ if and only if $G$ is a complete $k$-partite graph with
$k\geqslant 2$ and in this case the explicit formula for computation of
$\rho_{0,w}$ is obtained.
