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25
A geometrical approach to imprimitive graphs
 PROC. LONDON MATH. SOC
, 1995
"... We establish a geometrical framework for the study of imprimitive, Gsymmetric graphs F by exploiting the fact that any Gpartition 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) ..."
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Cited by 16 (3 self)
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We establish a geometrical framework for the study of imprimitive, Gsymmetric graphs F by exploiting the fact that any Gpartition 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 nondegenerate, a natural extremal case occurs when D(B) is a symmetric 2design with stabilizer G(B) acting doubly transitively on points: we characterize such graphs in the case where TB is complete.
Applications and adaptations of the low index subgroups procedure
 MATH. COMP
, 2005
"... The lowindex 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 parti ..."
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Cited by 7 (5 self)
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The lowindex 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.
Group Actions on Graphs, Maps and Surfaces with Maximum Symmetry
 in: Groups St Andrews 2001 in Oxford, London Math. Soc. Lecture Note Series
, 2003
"... 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 ni ..."
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Cited by 6 (3 self)
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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 sarctransitive graphs for large values of s. A brief description of the use of Schreier coset graphs and computational methods for handling nitelypresented groups and their images is also given.
Semisymetric graphs from polytopes
 J. Comb. Theory A 2007
"... Every ¯nite, selfdual, regular (or chiral) 4polytope of type f3; q; 3g has a trivalent 3transitive (or 2transitive) medial layer graph. Here, by dropping selfduality, we obtain a construction for semisymmetric trivalent graphs (which are edge but not vertextransitive). In particular, the Gra ..."
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Cited by 5 (2 self)
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Every ¯nite, selfdual, regular (or chiral) 4polytope of type f3; q; 3g has a trivalent 3transitive (or 2transitive) medial layer graph. Here, by dropping selfduality, we obtain a construction for semisymmetric trivalent graphs (which are edge but not vertextransitive). In particular, the Gray graph arises as the medial layer graph of a certain universal locally toroidal regular 4polytope. Key Words: semisymmetric graphs, abstract regular and chiral polytopes.
An infinite family of 4arctransitive cubic graphs each with girth 12
 Bull. London Math. Soc
, 1989
"... 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 regularl ..."
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Cited by 5 (3 self)
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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 4arcs. In fact if/? / 3 then this group is the full automorphism group of f(p), while the graph F(3) is 5arctransitive 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 4arcs. 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.
Finite Symmetric Graphs with TwoArc Transitive Quotients II
, 2006
"... DOI 10.1002/jgt.20260 Abstract: Let Ɣ be a finite Gsymmetric graph whose vertex set admits a nontrivial Ginvariant 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 articl ..."
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DOI 10.1002/jgt.20260 Abstract: Let Ɣ be a finite Gsymmetric graph whose vertex set admits a nontrivial Ginvariant 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 Gsymmetric 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 1design
On a class of finite symmetric graphs
 EUROPEAN JOURNAL OF COMBINATORICS
, 2007
"... Let Γ be a Gsymmetric graph, and let B be a nontrivial Ginvariant partition of the vertex set of Γ. This paper aims to characterize (Γ, G) under the conditions that the quotient graph Γ B is (G, 2)arc transitive and the induced subgraph between two adjacent blocks is 2 · K2 or K2,2. The results ..."
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Cited by 3 (2 self)
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Let Γ be a Gsymmetric graph, and let B be a nontrivial Ginvariant partition of the vertex set of Γ. This paper aims to characterize (Γ, G) under the conditions that the quotient graph Γ B is (G, 2)arc transitive and the induced subgraph between two adjacent blocks is 2 · K2 or K2,2. The results answer two questions about the relationship between Γ and Γ B for this class of graphs.
Classification of trivalent symmetric graphs of small order
 Australas. J. Combin
, 1995
"... A classification is given of all finite connected trivalent symmetric graphs on up to 240 vertices, based on an analysis of short relators in their automorphism groups. SUBJECT CLASSIFICATION 05C25(Primary) 20F05(Secondary) 1. ..."
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A classification is given of all finite connected trivalent symmetric graphs on up to 240 vertices, based on an analysis of short relators in their automorphism groups. SUBJECT CLASSIFICATION 05C25(Primary) 20F05(Secondary) 1.
On symmetries of Cayley graphs and the graphs underlying regular maps
"... By definition, Cayley graphs are vertextransitive, and graphs underlying regular or orientablyregular maps (on surfaces) are arctransitive. This paper addresses questions about how large the automorphism groups of such graphs can be. In particular, it is shown how to construct 3valent Cayley gra ..."
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By definition, Cayley graphs are vertextransitive, and graphs underlying regular or orientablyregular maps (on surfaces) are arctransitive. This paper addresses questions about how large the automorphism groups of such graphs can be. In particular, it is shown how to construct 3valent Cayley graphs that are 5arctransitive (in answer to a question by Cai Heng Li), and Cayley graphs of valency 3 t + 1 that are 7arctransitive, for all t> 0. The same approach can be taken in considering the graphs underlying regular or orientablyregular maps, leading to classifications of all such maps having a 1, 4 or 5arcregular 3valent underlying graph (in answer to questions by Cheryl Praeger and Sanming Zhou).