Related papers: Semiregular elements in cubic vertex-transitive gr…
Following Alspach and Parsons, a {\em metacirculant graph} is a graph admitting a transitive group generated by two automorphisms $\rho$ and $\sigma$, where $\rho$ is $(m,n)$-semiregular for some integers $m \geq 1$, $n \geq 2$, and where…
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…
We use the kernel category to give a finiteness condition for semigroups. As a consequence we provide yet another proof that finitely generated periodic semigroups of matrices are finite.
We show that the automorphism groups of right-angled Artin groups whose defining graphs have at least 3 vertices are not relatively hyperbolic. We then show that the outer automorphism groups are not relatively hyperbolic, if they are not…
A finite simple graph is called a bi-Cayley graph over a group $H$ if it has a semiregular automorphism group, isomorphic to $H,$ which has two orbits on the vertex set. Cubic vertex-transitive bi-Cayley graphs over abelian groups have been…
For a family $\mathcal{F}$ of graphs, let $ex(n,\mathcal{F})$ denote the maximum number of edges in an $n$-vertex graph which contains none of the members of $\mathcal{F}$ as a subgraph. A longstanding problem in extremal graph theory asks…
In the present paper we investigate the faithfulness of certain linear representations of groups of automorphisms of a graph $X$ in the group of symmetries of the Jacobian of $X$. As a consequence we show that if a $3$-edge-connected graph…
We present two results related to an edge-isoperimetric question for Cayley graphs on the integer lattice asked by Ben Barber and Joshua Erde [Isoperimetry of Integer Lattices, Discrete Analysis 7 (2018)]. For any (undirected) graph $G$,…
Half graphs and their variants, such as ladders, semi-ladders and co-matchings, are combinatorial objects that encode total orders in graphs. Works by Adler and Adler (Eur. J. Comb.; 2014) and Fabia\'nski et al. (STACS; 2019) prove that in…
The following problem is considered: if $H$ is a semiregular abelian subgroup of a transitive permutation group $G$ acting on a finite set $X$, find conditions for (non) existence of $G$-invariant partitions of $X$. Conditions presented in…
An embedding of a graph on an orientable surface is orientably-regular (or rotary, in an equivalent terminology) if the group of orientation-preserving automorphisms of the embedding is transitive (and hence regular) on incident vertex-edge…
We develop a new framework for analysing finite connected, oriented graphs of valency 4, which admit a vertex-transitive and edge-transitive group of automorphisms preserving the edge orientation. We identify a sub-family of "basic" graphs…
Given a graph $X$ with a Hamilton cycle $C$, the {\em compression factor $\kappa(X,C)$ of $C$} is the order of the largest cyclic subgroup of $\operatorname{Aut}(C)\cap\operatorname{Aut}(X)$, and the {\em Hamilton compression $\kappa(X)$ of…
We determine the maximum order of an element in the critical group of a strongly regular graph, and show that it achieves the spectral bound due to Lorenzini. We extend the result to all graphs with exactly two non-zero Laplacian…
A finite group $G$ admits an {\em oriented regular representation} if there exists a Cayley digraph of $G$ such that it has no digons and its automorphism group is isomorphic to $G$. Let $m$ be a positive integer. In this paper, we extend…
An infinite graph is quasi-transitive if its vertex set has finitely many orbits under the action of its automorphism group. In this paper we obtain a structure theorem for locally finite quasi-transitive graphs avoiding a minor, which is…
We show that one can define and effectively compute Stallings graphs for quasi-convex subgroups of automatic groups (\textit{e.g.} hyperbolic groups or right-angled Artin groups). These Stallings graphs are finite labeled graphs, which are…
We prove that, given a finite graph $\Sigma$ satisfying some mild conditions, there exist infinitely many tetravalent half-arc-transitive normal covers of $\Sigma$. Applying this result, we establish the existence of infinite families of…
In this paper, we explore a variety of finiteness questions for preperiodic points of morphisms. We begin by treating a group action analog of the Burnside problem for torsion groups using the p-adic arc method. We then prove some results…
Let $X$ be a finite vertex-transitive graph of valency $d$, and let $A$ be the full automorphism group of $X$. Then the arc-type of $X$ is defined in terms of the sizes of the orbits of the action of the stabiliser $A_v$ of a given vertex…