Related papers: Reconsideration of Tangle and Ultrafilter using Se…
We define the fundamental quandle of a spatial graph and several invariants derived from it. In the category of graph tangles, we define an invariant based on the walks in the graph and cocycles from nonabelian quandle cohomology.
Fractional graph isomorphism is the linear relaxation of an integer programming formulation of graph isomorphism. It preserves some invariants of graphs, like degree sequences and equitable partitions, but it does not preserve others like…
A divisibility relation on ultrafilters on the set $\mathbb{N}$ of natural numbers is defined as follows: ${\cal F}\hspace{1mm}\widetilde{\mid}\hspace{1mm}{\cal G}$ if and only if every set in $\cal F$ upward closed for divisibility also…
A graph parameter is self-dual in some class of graphs embeddable in some surface if its value does not change in the dual graph by more than a constant factor. We prove that the branchwidth of connected hypergraphs without bridges and…
Bifiltered graphs are a versatile tool for modelling relations between data points across multiple grades of a two-dimensional scale. They are especially popular in topological data analysis, where the homological properties of the induced…
In this work we introduce a convolution operation over the tangent bundle of Riemann manifolds in terms of exponentials of the Connection Laplacian operator. We define tangent bundle filters and tangent bundle neural networks (TNNs) based…
We define the tangential derivative, a notion of directional derivative which is invariant under diffeomorphisms. In particular this derivative is invariant under changes of chart and is thus well-defined for functions defined on a…
A tame filtration of an algebra is defined by the growth of its terms, which has to be majorated by an exponential function. A particular case is the degree filtration used in the definition of the growth of finitely generated algebras. The…
Recent times have seen a spurt of research activity focused on "completing" certain wave-particle duality relations using entanglement or polarization. These studies use a duality relation involving path-predictability, and not…
The bandwidth of a graph is the labeling of vertices with minimum maximum edge difference. For many graph families this is NP-complete. A classic result computes the bandwidth for the hypercube. We generalize this result to give sharp lower…
For a graph $G$ and partition $\mathcal{U}$ of its vertex set, an independent transversal of $(G, \mathcal{U})$ is an independent set of $G$ that contains one vertex from each block of $\mathcal{U}$. Buys, Kang, and Ozeki studied when a…
The fundamental role of on-shell diagrams in quantum field theory has been recently recognized. On-shell diagrams, or equivalently bipartite graphs, provide a natural bridge connecting gauge theory to powerful mathematical structures such…
Tangle structure trees, introduced in [3], offer a unified data structure that displays all the tangles of a graph or data set together with certificates for the non-existence of any other tangles, either locally or overall. In this paper…
Originally, tangles were invented as an abstract tool in mathematical graph theory to prove the famous graph minor theorem. In this paper, we showcase the practical potential of tangles in machine learning applications. Given a collection…
We introduce and study the Separation Problem for infinite graphs, which involves determining whether a connected graph splits into at least two infinite connected components after the removal of a given finite set of edges. We prove that…
Using tangent bundle geometry we construct an equivalent reformulation of classical field theory on flat spacetimes which simultaneously encodes the perspectives of multiple observers. Its generalization to curved spacetimes realizes a new…
We prove a duality theorem applicable to a a wide range of specialisations, as well as to some generalisations, of tangles in graphs. It generalises the classical tangle duality theorem of Robertson and Seymour, which says that every graph…
How might one "reduce" a graph? That is, generate a smaller graph that preserves the global structure at the expense of discarding local details? There has been extensive work on both graph sparsification (removing edges) and graph…
Bipartite graphs model the relationship between two disjoint sets of objects. They have a wide range of applications and are often visualized as a 2-layered drawing, where each set of objects is visualized as a set of vertices (points) on…
Network theory has proven to be a powerful tool in describing and analyzing systems by modelling the relations between their constituent objects. In recent years great progress has been made by augmenting `traditional' network theory.…