Angelo El Saliby | Searcharxiv

arXiv2026

Highly regular vertex-transitive graphs are globally rigid

A graph is said to be globally rigid in $d$-dimensional space if almost all of its embeddings are unique up to isometries. If a graph has enough automorphisms to send any of its vertices into any other, then it is called vertex-transitive. We show that, in any dimension, highly regular vertex-transitive graphs are globally rigid, positively answering a conjecture of Sean Dewar. Furthermore, we construct examples that show that our constant for regularity is best possible.

math.CO↗

arXiv2025

Computing Path Signature Varieties in Macaulay2

The signature of a path is a non-commutative power series whose coefficients are given by certain iterated integrals over the path coordinates. This series almost uniquely characterizes the path up to translation and reparameterization. Taking only fixed degree parts of these series yields signature tensors. We introduce the Macaulay2 package $\texttt{PathSignatures}$ to simplify the study of these interesting objects for piecewise polynomial paths. It allows for the creation and manipulation of parametrized families of paths and provides methods for computing their signature tensors and their associated algebraic varieties.

math.AG↗