Critical nets in $${\mathbb {R}}^k$$ R k (sometimes called geodesic nets) are embedded graph with the property that their embedding is a critical point of the total (edge) length functional and under the constraint that certain 1-valent vertices have a fixed position. In contrast to what happens on generic manifolds, we show that, if the embedding is bounded and n is the number of 1-valent vertices, the total length of the edges not incident with a 1-valent vertex is bounded by rn (where r is the outer radius), the degree of any vertex is bounded by n and that the number of edges (and hence the number of vertices) is bounded by $$n\ell $$ n ℓ (where $$\ell $$ ℓ is related to the combinatorial diameter of the graph).

中文翻译：

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在 $${\mathbb {R}}^k$$ 中的临界网络上

$${\mathbb {R}}^k$$ R k 中的临界网络（有时称为测地线网络）是嵌入图，其特性是它们的嵌入是总（边）长度泛函的临界点，并且在以下约束下某些一价顶点具有固定位置。与一般流形上发生的情况相反，我们表明，如果嵌入是有界的并且 n 是 1 价顶点的数量，则不与 1 价顶点相交的边的总长度以 rn 为界（其中 r是外半径），任何顶点的度数以 n 为界，边数（以及顶点数）以 $$n\ell $$ n ℓ 为界（其中 $$\ell $$ ℓ与图的组合直径有关）。