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16
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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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.
The finite vertexprimitive and vertexbiprimitive stransitive graphs with s ≥ 4
 Trans. Amer. Math. Soc
"... Abstract. A complete classification is given for finite vertexprimitive and vertexbiprimitive stransitive graphs for s ≥ 4. The classification involves the construction of new 4transitive graphs, namely a graph of valency 14 admitting the Monster simple group M, and an infinite family of graphs ..."
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Abstract. A complete classification is given for finite vertexprimitive and vertexbiprimitive stransitive graphs for s ≥ 4. The classification involves the construction of new 4transitive graphs, namely a graph of valency 14 admitting the Monster simple group M, and an infinite family of graphs of valency 5 admitting projective symplectic groups PSp(4,p)withp prime and p ≡±1 (mod 8). As a corollary of this classification, a conjecture of Biggs and Hoare (1983) is proved. 1.
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.
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 sarc transitive graphs of primepower order
 Bull. London Math. Soc
, 1993
"... An sarc in a graph is a vertex sequence (α0,α1,...,αs) such that {αi−1,αi} ∈EΓ for1�i � s and αi−1 � = αi+1 for 1 � i � s − 1. This paper gives a characterization of a class of stransitive graphs; that is, graphs for which the automorphism group is transitive on sarcs but not on (s + 1)arcs. It ..."
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An sarc in a graph is a vertex sequence (α0,α1,...,αs) such that {αi−1,αi} ∈EΓ for1�i � s and αi−1 � = αi+1 for 1 � i � s − 1. This paper gives a characterization of a class of stransitive graphs; that is, graphs for which the automorphism group is transitive on sarcs but not on (s + 1)arcs. It is proved that if Γ is a finite connected stransitive graph (where s � 2) of order a ppower with p prime, then s =2 or 3; further, either s = 3 and Γ is a normal cover of the complete bipartite graph K2m,2m,ors = 2 and Γ is a normal cover of one of the following 2transitive graphs: Kpm+1 (the complete graph of order pm+1), K2m,2m − 2mK2 (the complete bipartite graph of order 2m+1 minus a 1factor), a primitive affine graph, or a biprimitive affine graph. (Finite primitive and biprimitive affine 2arc transitive graphs were classified by Ivanov and Praeger in 1993.) 1.
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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Cited by 3 (1 self)
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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
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).
A more detailed classification of symmetric cubic graphs
"... A graph Γ is symmetric if its automorphism group acts transitively on the arcs of Γ, and sregular if its automorphism group acts regularly on the set of sarcs of Γ. Tutte (1947, 1959) showed that every cubic finite symmetric cubic graph is sregular for some s ≤ 5. Djokovič and Miller (1980) prove ..."
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A graph Γ is symmetric if its automorphism group acts transitively on the arcs of Γ, and sregular if its automorphism group acts regularly on the set of sarcs of Γ. Tutte (1947, 1959) showed that every cubic finite symmetric cubic graph is sregular for some s ≤ 5. Djokovič and Miller (1980) proved that there are seven types of arctransitive 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 arctransitive 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: Arctransitive graph, sregular graph, symmetric graph 2000 Mathematics Subject Classifications: 05C25, 20B25. 1
A NOTE ON GROUPS ASSOCIATED WITH 4ARCTRANSITIVE CUBIC GRAPHS
"... A cubic (trivalent) graph F is said to be 4arctransitive if its automorphism group acts transitively on the 4arcs of r (where a 4arc is a sequence «;0, vv...,vi of vertices of F such that t;,_j is adjacent to vt for 1 ^ I < 4, and vt1 ^ vi+1 for 1 < i < 4). In his investigations into g ..."
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A cubic (trivalent) graph F is said to be 4arctransitive if its automorphism group acts transitively on the 4arcs of r (where a 4arc is a sequence «;0, vv...,vi of vertices of F such that t;,_j is adjacent to vt for 1 ^ I < 4, and vt1 ^ vi+1 for 1 < i < 4). In his investigations into graphs of this sort, Biggs defined a family of groups 4 + (a m), for m = 3,4,5..., each presented in terms of generators and relations under the additional assumption that the vertices of a circuit of length m are cyclically permuted by some automorphism. In this paper it is shown that whenever m is a proper multiple of 6, the group 4 + (a m) is infinite. The proof is obtained by constructing transitive permutation representations of arbitrarily large degree. 1.