Can an arbitrary graph be embedded in Euclidean space so that the isometry group of its vertex set is precisely its graph automorphism group? This paper gives an affirmative answer, explores the number of dimensions necessary, and classifies the outerplanar graphs that have such an embedding in the plane.

Let s be a positive integer. A graph is s-transitive if its automorphism group is transitive on s-arcs but not on (s + 1)-arcs. Let p be a prime. In this article a complete classification of tetravalent s-transitive graphs of order 3p is given.

Abstract We prove that the automorphism group of semigeneric directed graph (in sense Cherlin’s classification) is uniquely ergodic.

A graph is vertex-transitive if its automorphism group acts transitively on its vertices. A vertex-transitive graph is a Cayley graph if its automorphism group contains a subgroup acting regularly on its vertices. In this paper, the cubic vertextransitive non-Cayley graphs of order 8p are classified for each prime p. It follows from this classification that there are two sporadic and two infini...

2010 Mathematics Subject Classifications: Primary 05C22; Secondary 05C15, 05C25, 05C30 Abstract. Up to switching isomorphism there are six ways to put signs on the edges of the Petersen graph. We prove this by computing switching invariants, especially frustration indices and frustration numbers, switching automorphism groups, chromatic numbers, and numbers of proper 1-colorations, thereby illu...

We prove that every simplicial automorphism of the free splitting graph of a free group Fn is induced by an outer automorphism of Fn for n ≥ 3. In this note we consider the graph Gn of free splittings of the free group Fn of rank n ≥ 3. Loosely speaking, Gn is the graph whose vertices are non-trivial free splittings of Fn up to conjugacy, and where two vertices are adjacent if they are represen...

Computational techniques are described for the automorphism groups of edge-weighted graphs. Fortran codes based on the manipulation of weighted adjacency matrices are used to compute the automorphism groups of several edge-weighted graphs. The code developed here took 37l/2 min of CPU time to generate 1 036 800 permutations in the automorphism group of an edge-weighted graph.

The polycirculant conjecture asserts that every vertex-transitive digraph has a semiregular automorphism, that is, a nontrivial automorphism whose cycles all have the same length. In this paper we investigate the existence of semiregular automorphisms of edge-transitive graphs. In particular, we show that any regular edge-transitive graph of valency three or four has a semiregular automorphism.

A graph G is called minimalizable if a diagram with minimal crossing number can be obtained from an arbitrary diagram of G by crossing changes. If, furthermore, the minimal diagram is unique up to crossing changes then G is called strongly minimalizable. In this article, it is explained how minimalizability of a graph is related to its automorphism group and it is shown that a graph is strongly...

One of the most striking impacts between geometry, combinatorics and graph theory, on one hand, and algebra and group theory, on the other hand, arise from a concrete necessity to manipulate with the symmetry of the investigated objects. In the case of graphs, we talk about such tasks as identification and compact representation of graphs, recognition of isomorphic graphs and computation of aut...