Quantum automorphism groups of homogeneous graphs

Associated to a finite graph X is its quantum automorphism group G. The main problem is to compute the Poincaré series of G, meaning the series f(z)=1+c₁z+c₂z²+? whose coefficients are multiplicities of 1 into tensor powers of the fundamental representation. In this paper we find a duality between certain quantum groups and planar algebras, which leads to a planar algebra formulation of the problem. Together with some other results, this gives f for all homogeneous graphs having 8 vertices or less.     

Geometric automorphism groups of graphs

Constructing symmetric drawings of graphs is NP-hard. In this paper, we present a new method for drawing graphs symmetrically based on group theory. More formally, we define an n-geometric automorphism group as a subgroup of the automorphism group of a graph that can be displayed as symmetries of a drawing of the graph in n dimensions. Then we present an algorithm to find all 2- and 3-geometric automorphism groups of a given graph. We implement the algorithm using Magma [〈http://magma.maths.usyd.edu.au〉] and the experimental results show that our approach is very efficient in practice. We also present a drawing algorithm to display 2- and 3-geometric automorphism groups.     

A note on the automorphism groups of cubic Cayley graphs of finite simple groups

For a finite group G, a Cayley graph on G is said to be normal if . In this note, we prove that connected cubic non-symmetric Cayley graphs of the ten finite non-abelian simple groups G in the list of non-normal candidates given in [X.G. Fang, C.H. Li, J. Wang, M.Y. Xu, On cubic Cayley graphs of finite simple groups, Discrete Math. 244 (2002) 67-75] are normal.     

On automorphism groups of Cayley graphs

LetX _G,H denote the Cayley graph of a finite groupG with respect to a subsetH. It is well-known that its automorphism groupA(X_G,H) must contain the regular subgroupL _G corresponding to the set of left multiplications by elements ofG. This paper is concerned with minimizing the index [A(X_G,H)L_G] for givenG, in particular when this index is always greater than 1. IfG is abelian but not one of seven exceptional groups, then a Cayley graph ofG exists for which this index is at most 2. Nearly complete results for the generalized dicyclic groups are also obtained.     

On automorphism groups of some finite groups

We show that if n > 1 is odd and has no divisor p4 for any prime p, then there is no finite group G such that│Aut(G)│ = n.     

Direct products of automorphism groups of graphs

In this article we study the product action of the direct product of automorphism groups of graphs. We generalize the results of Watkins [J. Combin Theory 11 (1971), 95–104], Nowitz and Watkins [Monatsh. Math. 76 (1972), 168–171] and W. Imrich [Israel J. Math. 11 (1972), 258–264], and we show that except for an infinite family of groups S_n × S_n, n≥2 and three other groups D₄ × S₂, D₄ × D₄ and S₄ × S₂ × S₂, the direct product of automorphism groups of two graphs is itself the automorphism group of a graph. © 2009 Wiley Periodicals, Inc. J Graph Theory 62: 26–36, 2009     

On automorphism groups of some PCS graphs

In this paper we review the characterization of point-color symmetric (PCS) graphs based on the color preserving automorphisms given in [3]. In particular, we consider PCS pictures, arriving at another characterization theorem. We summarize a few results and give some examples.     

On coleman outer automorphism groups of finite groups

Let G be a finite group and Out _Col(G) the Coleman outer automorphism group of G(for the definition, see below). The question whether Out_Col(G) is a p′-group naturally arises from the study of the normalizer problem for integral group rings, where p is a prime. In this article, some sufficient conditions for Out_Col(G) to be a p′-group are obtained. Our results generalize some well-known theorems.     

Symmetric cubic graphs with solvable automorphism groups

A cubic graph $\Gamma$ is $G$-arc-transitive if $G\le \mathrm{Aut}\left(\Gamma \right)$ acts transitively on the set of arcs of $\Gamma$, and $G$-basic if $\Gamma$ is $G$-arc-transitive and $G$ has no non-trivial normal subgroup with more than two orbits. Let $G$ be a solvable group. In this paper, we first classify all connected $G$-basic cubic graphs and determine the group structure for every $G$. Then, combining covering techniques, we prove that a connected cubic $G$-arc-transitive graph is either a Cayley graph, or its full automorphism group is of type $2^2$, that is, a $2$-regular group with no involution reversing an edge, and has a non-abelian normal subgroup such that the corresponding quotient graph is the complete bipartite graph of order $6$.     

Holonomy groups of Bieberbach groups with finite outer automorphism groups

This work was supported by the DFG and by a Polish grant BW/5100-5-0116-3.     

On automorphism groups of graphs and distributive lattices

Birkhoff’s theorem that every group is isomorphic to the automorphism group of a distributive lattice is extended in a direction that parallels similar results in graph theory. Received December 9, 1997; accepted in final form October 9, 1998.     

On automorphism groups of quasiprimitive 2-arc transitive graphs

We characterize the automorphism groups of quasiprimitive 2-arc-transitive graphs of twisted wreath product type. This is a partial solution for a problem of Praeger regarding quasiprimitive 2-arc transitive graphs. The solution stimulates several further research problems regarding automorphism groups of edge-transitive Cayley graphs and digraphs. This work forms part of an ARC grant project and is supported by a QEII Fellowship.     

Solvable line-transitive automorphism groups of finite linear spaces

Let S be a finite linear space, and letG be a group of automorphisms of S. IfG is soluble and line-transitive, then for a givenk but a finite number of pairs of (S, G),S hasv= p ⁿ points andG ⩽AΓL(1,p ⁿ ).