Related papers: Twists and Twistability
Let $X=(V\!X,E\!X)$ be an infinite, locally finite, connected graph without loops or multiple edges. We consider the edges to be oriented, and $E\!X$ is equipped with an involution which inverts the orientation. Each oriented edge is…
We propose a generalisation of the notion of associated bundles to a principal bundle constructed via group action cocycles rather than via mere representations of the structure group. We devise a notion of connection generalising Ehresmann…
We consider the infinite-dimensional hypercube graph. This graph is not connected and has isomorphic connected components. We describe the restrictions of its automorphisms to the connected components and the automorphism group of connected…
An arbitrary Feynman graph for string field theory interactions is analysed and the homeomorphism type of the corresponding world sheet surface is completely determined even in the non-orientable cases. Algorithms are found to mechanically…
Any representation of data involves arbitrary investigator choices. Because those choices are external to the data-generating process, each choice leads to an exact symmetry, corresponding to the group of transformations that takes one…
The second author showed how Katsura's construction of the C*-algebra of a topological graph E may be twisted by a Hermitian line bundle L over the edge space E. The correspondence defining the algebra is obtained as the completion of the…
We introduce AutoGraph, a scalable autoregressive model for attributed graph generation using decoder-only transformers. By flattening graphs into random sequences of tokens through a reversible process, AutoGraph enables modeling graphs as…
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…
We classify the connected-homogeneous digraphs with more than one end. We further show that if their underlying undirected graph is not connected-homogeneous, they are highly-arc-transitive.
This article studies automorphism groups of graph products of arbitrary groups. We completely characterise automorphisms that preserve the set of conjugacy classes of vertex groups as those automorphisms that can be decomposed as a product…
A map on a group into itself is called a local automorphism if at any two points of the group, it can be interpolated by an automorphism of that group. In this paper we investigate the question of how local automorphisms of some classical…
A relational structure is homomorphism-homogeneous if every homomorphism between finite substructures extends to an endomorphism of the structure. This notion was introduced recently by Cameron and Ne\v{s}et\v{r}il. In this paper we…
We introduce a topological approach to words. Words are approximated by Gauss words and then studied up to natural modifications inspired by homotopy transformations of curves on the plane.
Graph classification plays an important role is data mining, and various methods have been developed recently for classifying graphs. In this paper, we propose a novel method for graph classification that is based on homotopy equivalence of…
Homotopy Lie algebras are a generalization of differential graded Lie algebras encoding both the kinematics and dynamics of a given field theory. Focusing on kinematics, we show that these algebras provide a natural framework for the…
In this article, we study and review some aspects of twisted cohomologies on algebraic and analytic varieties. We compared such cohomologies in both the algebraic and analytic categories and defined two types of twisting parameters in the…
An automorphism of a graph $G=(V,E)$ is a bijective map $\phi$ from $V$ to itself such that $\phi(v_i)\phi(v_j)\in E$ $\Leftrightarrow$ $v_i v_j\in E$ for any two vertices $v_i$ and $v_j$. Denote by $\mathfrak{G}$ the group consisting of…
We consider a variety of connections between threshold graphs, shifted complexes, and simplicial complexes naturally formed from a graph. These graphical complexes include the independent set, neighborhood, and dominance complexes. We…
The automorphisms of a graph act naturally on its set of labeled imbeddings to produce its unlabeled imbeddings. The imbedding sum of a graph is a polynomial that contains useful information about a graph's labeled and unlabeled imbeddings.…
After the introduction of $\lambda$-symmetries by Muriel and Romero, several other types of so called "twisted symmetries" have been considered in the literature (their name refers to the fact they are defined through a deformation of the…