On Critical nets in $\mathbb{R}^k$

TL;DR

This paper investigates the properties of critical geodesic nets in Euclidean space, establishing bounds on their total length, degrees, and size based on the number of leaves and geometric constraints.

Contribution

It provides new bounds on the total length, vertex degree, and size of critical nets in Euclidean space, linking geometric and combinatorial properties.

Findings

Total length of edges not incident to leaves is bounded by rn.

Vertex degree is bounded by the number of leaves n.

Number of edges and vertices is bounded by nl, related to the graph's diameter.

Abstract

Critical nets in $\mathbb{R}^k$ (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 (leaves) have a fixed position. In contrast to what happens on generic manifolds, we show that, if 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 nl where l is related to the combinatorial diameter of the graph.