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We introduce techniques for proving uniform termination of graph transformation systems, based on matrix interpretations for string rewriting. We generalize this technique by adapting it to graph rewriting instead of string rewriting and by…
We consider a finite, connected and simple graph $\Gamma$ that admits a vertex-transitive group of automorphisms $G$. Under the assumption that, for all $x \in V(\Gamma)$, the local action $G_x^{\Gamma(x)}$ is the action of…
We develop a theory of $\times$-homotopy, fundamental groupoids and covering spaces that apply to non-simple graphs, generalizing existing results for simple graphs. We prove that $\times$-homotopies from finite graphs can be decomposed…
A restricted permutation of a locally finite directed graph $G=(V,E)$ is a vertex permutation $\pi: V\to V$ for which $(v,\pi(v))\in E$, for any vertex $v\in V$. The set of such permutations, denoted by $\Omega(G)$, with a group action…
The `random intersection graph with communities' models networks with communities, assuming an underlying bipartite structure of groups and individuals. Each group has its own internal structure described by a (small) graph, while groups…
Graph transformation theory relies upon the composition of rules to express the effects of sequences of rules. In practice, graphs are often subject to constraints, ruling out many candidates for composed rules. Focusing on the case of…
Let $G=C_{p^n}$ be a finite cyclic p-group, and let $Hol(G)$ denote its holomorph. In this work, we find and characterize the regular subgroups of $Hol(G)$ that are mutually normalizing each other in the permutation group $Sym(G)$. We…
Various connections between the theory of permutation groups and the theory of topological groups are described. These connections are applied in permutation group theory and in the structure theory of topological groups. The first draft of…
We describe a technique to determine the automorphism group of a geometrically represented graph, by understanding the structure of the induced action on all geometric representations. Using this, we characterize automorphism groups of…
A rank 3 graph is an orbital graph of a rank 3 permutation group of even order. Despite the classification of rank 3 graphs being complete, see, e.g., Chapter 11 of the recent monograph 'Strongly regular graphs' by Brouwer and Van…
We show that all permutations in $S_n$ can be generated by affine unicritical polynomials. We use the $\operatorname{PGL}$ group structure to compute the cycle structure of permutations with low Carlitz rank. The tree structure of the group…
Probabilistic graphical models with frustration exhibit rugged energy landscapes that trap iterative optimization dynamics. These landscapes are shaped not only by local interactions, but crucially also by the global loop structure of the…
This paper introduces a group-theoretic framework to analyze the algebraic structure of the Grover walk on a complete graph with self-loops. We construct a group generated by the Grover matrix and a diagonal matrix whose entries are powers…
We introduce a graphical representation of stabilizer states and translate the action of Clifford operators on stabilizer states into graph operations on the corresponding stabilizer-state graphs. Our stabilizer graphs are constructed of…
Motivated by the problem of deformation quantization we introduce and study directed graph complexes with oriented loops and wheels. We develop some technique for computing cohomology of such graph complexes and apply it to several concrete…
Let $\Gamma$ be a simple undirected graph on a finite vertex set and let $A$ be its adjacency matrix. Then $\Gamma$ is {\it singular} if $A$ is singular. The problem of characterising singular graphs is easy to state but very difficult to…
Let $X=(V,E)$ be a finite simple connected graph with $n$ vertices and $m$ edges. A configuration is an assignment of one of two colors, black or white, to each edge of $X.$ A move applied to a configuration is to select a black edge…
Comparability graphs are graphs which have transitive orientations. The dimension of a poset is the least number of linear orders whose intersection gives this poset. The dimension ${\rm dim}(X)$ of a comparability graph $X$ is the…
The operation of transforming one spanning tree into another by replacing an edge has been considered widely, both for general and planar straight-line graphs. For the latter, several variants have been studied (e.g., edge slides and edge…
We introduce the \emph{intersection orbital graph} $\Gamma(G_1, G_2; \Omega)$ associated with two permutation groups $G_1, G_2 \leq \mathrm{Sym}(\Omega)$ on a finite set $\Omega$.