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Abstract. The low-index subgroups procedure is an algorithm for finding all subgroups of up to a given index in a finitely presented group G and hence for determining all transitive permutation representations of G of small degree. A number of significant applications of this algorithm are discussed, in particular to the construction of graphs and surfaces with large automorphism groups. Furthermore, three useful adaptations of the procedure are described, along with parallelisation of the algorithm. In particular, one adaptation finds all complements of a given finite subgroup (in certain contexts), and another finds all normal subgroups of small index in the group G. Significant recent applications of these are also described in some detail. 1.

We establish a geometrical framework for the study of imprimitive, G-symmetric graphs F by exploiting the fact that any G-partition B of the vertex set VT gives rise both to a quotient graph fB and to a tactical configuration D(B) induced on each block BeB. We also examine those cases in which D(B) is degenerate, and characterize the possible graphs f in many cases where the quotient FB is either a complete graph or a circuit. When D(fl) is non-degenerate, a natural extremal case occurs when D(B) is a symmetric 2-design with stabilizer G(B) acting doubly transitively on points: we characterize such graphs in the case where TB is complete. 1.

This is a summary of a short course of lectures given at the Groups St Andrews conference in Oxford, August 2001, on the signi cant role of combinatorial group theory in the study of objects possessing a high degree of symmetry. Topics include group actions on closed surfaces, regular maps, and nite s-arc-transitive graphs for large values of s. A brief description of the use of Schreier coset graphs and computational methods for handling nitely-presented groups and their images is also given.

If p is any prime, and 6 is that automorphism of the group SL(3,/>) which takes each matrix to the transpose of its inverse, then there exists a connected trivalent graph F(p) on ^p 3 (p 3 —\)(p 2 — \) vertices with the split extension SL(3,/?)<0> as a group of automorphisms acting regularly on its 4-arcs. In fact if/? / 3 then this group is the full automorphism group of f(p), while the graph F(3) is 5-arc-transitive with full automorphism group SL(3,3)<0> x C2. The girth of F(p) is 12, except in the case p = 2 (where the girth is 6). Furthermore, in all cases F(p) is bipartite, with SL(3,p) fixing each part. Also when p = 1 mod 3 the graph T(p) is a triple cover of another trivalent graph, which has automorphism group PSL(3,p)<0> acting regularly on its 4-arcs. These claims are proved using elementary theory of symmetric graphs, together with a suitable choice of three matrices which generate SL(3, Z). They also provide a proof that the group 4 + (a 12) described by Biggs in Computational group theory (ed. M. Atkinson) is infinite.

Every finite, self-dual, regular (or chiral) 4-polytope of type {3, q,3} has a trivalent 3-transitive (or 2-transitive) medial layer graph. Here, by dropping self-duality, we obtain a construction for semisymmetric trivalent graphs (which are edge- but not vertex-transitive). In particular, the Gray graph arises as the medial layer graph of a certain universal locally toroidal regular 4-polytope. Key Words: semisymmetric graphs, abstract regular and chiral polytopes.

By definition, Cayley graphs are vertex-transitive, and graphs underlying regular or orientably-regular maps (on surfaces) are arc-transitive. This paper addresses questions about how large the automorphism groups of such graphs can be. In particular, it is shown how to construct 3-valent Cayley graphs that are 5-arc-transitive (in answer to a question by Cai Heng Li), and Cayley graphs of valency 3 t + 1 that are 7-arc-transitive, for all t> 0. The same approach can be taken in considering the graphs underlying regular or orientably-regular maps, leading to classifications of all such maps having a 1-, 4- or 5-arc-regular 3-valent underlying graph (in answer to questions by Cheryl Praeger and Sanming Zhou).

A graph Γ is symmetric if its automorphism group acts transitively on the arcs of Γ, and s-regular if its automorphism group acts regularly on the set of s-arcs of Γ. Tutte (1947, 1959) showed that every cubic finite symmetric cubic graph is s-regular for some s ≤ 5. Djokovič and Miller (1980) proved that there are seven types of arc-transitive group action on finite cubic graphs, characterised by the stabilisers of a vertex and an edge. A given finite symmetric cubic graph, however, may admit more than one type of arc-transitive group action. In this paper we determine exactly which combinations of types are possible. Some combinations are easily eliminated by existing theory, and others can be eliminated by elementary extensions of that theory. The remaining combinations give 17 classes of finite symmetric cubic graph, and for each of these, we prove the class is infinite, and determine at least one representative. For at least 14 of these 17 classes the representative we give has the minimum possible number of vertices (and we show that in two of these 14 cases every graph in the class is a cover of the smallest representative), while for the other three classes, we give the smallest examples known to us. In an Appendix, we give a table showing the class of every symmetric cubic graph on up to 768 vertices. Keywords: Arc-transitive graph, s-regular graph, symmetric graph 2000 Mathematics Subject Classifications: 05C25, 20B25. 1

DOI 10.1002/jgt.20260 Abstract: Let Ɣ be a finite G-symmetric graph whose vertex set admits a nontrivial G-invariant partition B. It was observed that the quotient graph ƔB of Ɣ relative to B can be (G, 2)-arc transitive even if Ɣ itself is not necessarily (G, 2)-arc transitive. In a previous article of Iranmanesh et al., this observation motivated a study of G-symmetric graphs (Ɣ, B) such that ƔB is (G, 2)-arc transitive and, for blocks B, C ∈ B adjacent in ƔB, there are exactly |B|−2(≥1) vertices in B which have neighbors in C. In the present article we investigate the general case where ƔB is (G, 2)-arc transitive and is not multicovered by Ɣ (i.e., at least one vertex in B has no neighbor in C for adjacent B, C ∈ B) by analyzing the dual D ∗ (B) of the 1-design

We survey computers systems which help to obtain and sometimes provide automatically conjectures and refutations in algebraic graph theory.