Related papers: Grid homology for spatial graphs and a skein exact…
Graph theory provides a language for studying the structure of relations, and it is often used to study interactions over time too. However, it poorly captures the both temporal and structural nature of interactions, that calls for a…
The Image-Computing Spectral Sequence computes the homology of the image of a finite map from the alternating homology of the multiple point spaces of the map. A related spectral sequence was obtained by Gabrielov, Vorobjob and Zell which…
We investigate the structure of graphs of twin-width at most $1$, and obtain the following results: - Graphs of twin-width at most $1$ are permutation graphs. In particular they have an intersection model and a linear structure. - There is…
We define a chain complex for generalized splines on graphs, analogous to that introduced by Billera and refined by Schenck-Stillman for splines on polyhedral complexes. The hyperhomology of this chain complex yields bounds on the…
Recent years have witnessed fast developments of graph neural networks (GNNs) that have benefited myriad graph analytic tasks and applications. Most GNNs rely on the homophily assumption that nodes belonging to the same class are more…
In a previous paper, V\'ertesi and the first author used grid-like Heegaard diagrams to define tangle Floer homology, which associates to a tangle $T$ a differential graded bimodule $\widetilde{\mathrm{CT}} (T)$. If $L$ is obtained by…
Homology of braid groups and Artin groups can be related to the study of spaces of curves. We completely calculate the integral homology of the family of smooth curves of genus $g$ with one boundary component, that are double coverings of…
We propose to study homomorphisms of connectome graphs. Homomorphisms can be studied as sequences of elementary homomorphisms - folds, which identify pairs of vertices. Several fold types are defined. Initial computation results for some…
Geometric representations of cycles in quandle homology theory are given in terms of colored knot diagrams. Abstract knot diagrams are generalized to diagrams with exceptional points which, when colored, correspond to degenerate cycles.…
Spin networks, essentially labeled graphs, are ``good quantum numbers'' for the quantum theory of geometry. These structures encompass a diverse range of techniques which may be used in the quantum mechanics of finite dimensional systems,…
The purpose of this note is to extend to Brownian loops some homology and holonomy results obtained in the case of discrete loops on a graph
In 2010, Turaev introduced knotoids as a variation on knots that replaces the embedding of a circle with the embedding of a closed interval with two endpoints which here we call poles. We define generalized knotoids to allow arbitrarily…
We present a novel theoretical framework connecting k-component edge connectivity with spectral graph theory and homology theory to pro vide new insights into the resilience of real-world networks. By extending classical edge connectivity…
In this paper, we study Dehn colorings of spatial graph diagrams, and classify the vertex conditions, equivalently the palettes. We give some example of spatial graphs which can be distinguished by the number of Dehn colorings with…
We introduce several commutative rings, the snake rings, that have strong connections to cluster algebras. The elements of these rings are residue classes of unions of certain labeled graphs that were used to construct canonical bases in…
In real-world systems, the relationships and connections between components are highly complex. Real systems are often described as networks, where nodes represent objects in the system and edges represent relationships or connections…
Given a graph $\Gamma$, one may conside the set $X$ of its vertices as a metric space by assuming that all edges have length one. We consider two versions of homology theory of $\Gamma$ and their $K$-theory counterparts -- the $K$-theory of…
This paper presents several algorithms for hashing directed graphs. The algorithms given are capable of hashing entire graphs as well as assigning hash values to specific nodes in a given graph. The notion of node symmetry is made precise…
This paper focuses on developing an efficient algorithm for analyzing a directed network (graph) from a topological viewpoint. A prevalent technique for such topological analysis involves computation of homology groups and their…
We study the logarithmic topological Hochschild homology of ring spectra with logarithmic structures and establish localization sequences for this theory. Our results apply, for example, to connective covers of periodic ring spectra like…