A graph X is said to be 1 2-transitive if its automorphism group Aut X acts vertex-and edge-but not arc-transitively on X. Then Aut X induces an orientation of the edges of X. If X has valency 4, then this orientation gives rise to so called alternating cycles, that is even length cycles in X whose every other vertex is the head and every other vertex is the tail of its two incident edges in th...

We show that 3 is the smallest order of a non-trivial odd order group which occurs as the full automorphism group of a finite group.

In part I, we showed that there was an infinite family of graphs not embeddable in the torus in such a way that a free action of any group on any of the graphs extends to a cellular automorphism of the torus. We used a homologically-driven analysis of ordinary voltage graph embeddings. An ordinary voltage graph embedding with voltage group A in a surface encodes a derived embedding of a derived...

As it is introduced by Bermond, Prennes, and Kodate and by Fragopoulou and Akl, some Cayley graphs, including most popular models for intercon-nection networks, admit a special automorphism, called complete rotation. Such an automorphism is often used to derive algorithms or properties of the underlying graph. For example, some optimal gossiping algorithms can be easily designed by using a comp...

We study automorphism groups of sparse graphs from the viewpoint of topological dynamics and the Kechris, Pestov, Todorčević correspondence. We investigate amenable and extremely amenable subgroups of these groups using the space of orientations of the graph and results from structural Ramsey theory. Resolving one of the open questions in the area, we show that Hrushovski’s example of an ω-cate...

we prove that each normal automorphism of the $n$-periodic product of cyclic groups of odd order $rge1003$ is inner, whenever $r$ divides $n$.

We show that the outer automorphism groups of graph products finitely generated abelian satisfy Tits alternative, are residually finite, their so-called Torelli subgroups generated, and they a dichotomy between being virtually nilpotent containing non-abelian free subgroup is determined by graphical condition on underlying labelled graph. Graph simultaneously generalize right-angled Artin (RAAG...

In this paper, various algorithms used in the classifications of regular polytopes for given groups are compared. First computational times and memory usages are analyzed for the original algorithm used in one of these classifications. Second, a possible algorithm for isomorphism testing among polytopes is suggested. Then, two improved algorithms are compared, and finally, results are given for...

The fixing number of a graph G is the smallest cardinality of a set of vertices S such that only the trivial automorphism of G fixes every vertex in S. The fixing set of a group Γ is the set of all fixing numbers of finite graphs with automorphism group Γ. Several authors have studied the distinguishing number of a graph, the smallest number of labels needed to label G so that the automorphism ...

The operation of switching a graph Γ with respect to a subset X of the vertex set interchanges edges and non-edges between X and its complement, leaving the rest of the graph unchanged. This is an equivalence relation on the set of graphs on a given vertex set, so we can talk about the automorphism group of a switching class of graphs. It might be thought that switching classes with many automo...