Abstract: Let ${\cal{G}} = (V, E, d)$ be any connected weighted graph which admits not only degenerated realisations in the $n$-dimensional Euclidean space. Its configuration space is always homeomorphic to a $({1\over 2} n(n+1) - e)$-dimensional sphere, where $n$ is the number of vertices minus one and $e$ the number of edges.
Keywords: weighted graph; realisations in $\bbf R^n$; distance-preserving embeddings of graphs in Euclidean space
Classification (MSC2000): 52A37; 05C62
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