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Among other results, we prove the following theorem about Steiner minimal trees in $d$-dimensional Euclidean space: if two finite sets in $mathbb{R}^d$ have unique and combinatorially equivalent Steiner minimal trees, then there is a homotopy between the two sets that maintains the uniqueness and the combinatorial structure of the Steiner minimal tree throughout the homotopy.
In the classical Steiner tree problem, given an undirected, connected graph $G=(V,E)$ with non-negative edge costs and a set of emph{terminals} $Tsubseteq V$, the objective is to find a minimum-cost tree $E subseteq E$ that spans the terminals. The p
We study the Lipschitz structures on the geodesic compactification of a regular tree, that are preserved by the automorphism group. They are shown to be similar to the compactifications introduced by William Floyd, and a complete description is given.
We study configuration space integral formulas for Milnors homotopy link invariants, showing that they are in correspondence with certain linear combinations of trivalent trees. Our proof is essentially a combinatorial analysis of a certain space of
Graph neural networks have been successful in many learning problems and real-world applications. A recent line of research explores the power of graph neural networks to solve combinatorial and graph algorithmic problems such as subgraph isomorphism
In this paper we prove upper and lower bounds on the minimal spherical dispersion. In particular, we see that the inverse $N(varepsilon,d)$ of the minimal spherical dispersion is, for fixed $varepsilon>0$, up to logarithmic terms linear in the dimens