Geometric realization of graph

Geometric realization of graph] --- геометрическая реализация графа.

Let [math]\displaystyle{ G = (V,E) }[/math] be an undirected graph with weights [math]\displaystyle{ 1/c_{e} }[/math] for each edge [math]\displaystyle{ \in E }[/math]. The geometric realization of [math]\displaystyle{ G }[/math] is the metric space [math]\displaystyle{ {\mathcal G} }[/math] consisting of [math]\displaystyle{ V }[/math] and arcs of length [math]\displaystyle{ c_{e} }[/math] glued between [math]\displaystyle{ u }[/math] and [math]\displaystyle{ v }[/math] for each edge [math]\displaystyle{ e = (u,v) \in E }[/math]. The volume [math]\displaystyle{ \mu(G) }[/math] is the Lebesgue measure of [math]\displaystyle{ {\mathcal G} }[/math], i.e.

[math]\displaystyle{ \mu(G) = \sum_{e \in E} c_{e}. }[/math]