Related papers: An elementary proof of a conjecture on graph-autom…

We derive a formula connecting the orders of the automorphism groups of a finite group and of its covering groups.

Beck's conjecture on coloring of graphs associated to various algebraic objects has generated considerable interest in the community of discrete mathematics and combinatorics since its inception in the year 1988. The version of this…

We formulate a categorification of Robertson's conjecture analogous to the categorical graph minor conjecture of Miyata--Proudfood--Ramos. We show that these conjectures imply the existence of a finite list of atomic graphs generating the…

We give upper bounds on the order of the automorphism group of a simple graph

We introduce a recursive method to deconstruct the automorphism group of an ordered set. By connecting this method with deep results for permutation groups, we prove the Automorphism Conjecture for ordered sets of width less than or equal…

We prove that every finite group is the automorphism group of a finite abstract polytope isomorphic to a face-to-face tessellation of a sphere by topological copies of convex polytopes. We also show that this abstract polytope may be…

Using combinatorial techniques, we prove that the weighted sum of the inverse number of automorphisms of all finite abelian $p$-groups $\sum_G |G|^{-u} |\text{Aut}(G)|^{-1}$ is equal to $\prod_{j=u+1}^\infty\left(1-1/p^j\right)^{-1}$, where…

Polytopes are ubiquitous in different areas of mathematics. Gleason and Hubard established a factorisation theorem, stating that every abstract polytope has a unique factorisation into prime polytopes. We compute the automorphism group of a…

A combinatorial map is a connected topological graph cellularly embedded in a surface. This monograph concentrates on the automorphism group of a map, which is related to the automorphism group of a Klein surface and a Smarandache manifold,…

A trivial automorphism of the Boolean algebra $\mathcal P(\mathbb N) / \mathrm{Fin}$ is an automorphism induced by the action of some function $\mathbb N \rightarrow \mathbb N$. In models of forcing axioms all automorphisms are trivial, and…

We prove that, to every abstract group $G$, we can associate a sequence of graphs $\Gamma_n$ such that the automorphism group of $\Gamma_n$ is isomorphic to $G$ and the genus of $\Gamma_n$ is an unbounded function of $n$.

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…

In 2007, Dmytrenko, Lazebnik and Williford posed two related conjectures about polynomials over finite fields. Conjecture~1 is a claim about the uniqueness of certain monomial graphs. Conjecture~2, which implies Conjecture~1, deals with…

J.H.C. Whitehead's second free-group algorithm determines whether or not two given elements of a free group lie in the same orbit of the automorphism group of the free group. The algorithm involves certain connected graphs, and Whitehead…

Automorphism groups are intrincate conjugacy invariants for subshifts, which can reveal important features of the dynamical structure of a shift action. One important case is the study of automorphism groups when the underlying subshift has…

There are a variety of ways to associate directed or undirected graphs to a group. It may be interesting to investigate the relations between the structure of these graphs and characterizing certain properties of the group in terms of some…

We give a simpler proof using automata theory of a recent result of Kapovich, Weidmann and Myasnikov according to which so-called benign graphs of groups preserve decidability of the generalized word problem. These include graphs of groups…

In this article we study the K- and L-theory of groups acting on trees. We consider the problem in the context of the fibered isomorphism conjecture of Farrell and Jones. We show that in the class of residually finite groups it is enough to…

We determine the structure of automorphism groups of finite graphs of bounded Hadwiger number. Our proof includes a structural analysis of finite edge-transitive graphs. In particular, we show that for connected, $K_{h+1}$-minor-free,…

The non--commuting graph $\Gamma(G)$ of a non--abelian group $G$ is defined as follows. The vertex set $V(\Gamma(G))$ of $\Gamma(G)$ is $G\setminus Z(G)$ where $Z(G)$ denotes the center of $G$ and two vertices $x$ and $y$ are adjacent if…