On cubic symmetric non-Cayley graphs with solvable automorphism groups (original) (raw)

Automorphisms of a Family of Cubic Graphs

Algebra Colloquium, 2013

A Cayley graph Cay (G,S) on a group G with respect to a Cayley subset S is said to be normal if the right regular representation R(G) of G is normal in the full automorphism group of Cay (G,S). For a positive integer n, let Γn be a graph with vertex set {xi,yi|i ∈ ℤ2n} and edge set {{xi,xi+1}, {yi,yi+1}, {x2i,y2i+1}, {y2i,x2i+1}|i ∈ ℤ2n}. In this paper, it is shown that Γn is a Cayley graph and its full automorphism group is isomorphic to [Formula: see text] for n=2, and to [Formula: see text] for n > 2. Furthermore, we determine all pairs of G and S such that Γn= Cay (G,S) is non-normal for G. Using this, all connected cubic non-normal Cayley graphs of order 8p are constructed explicitly for each prime p.

On the Automorphism Groups of Cayley Graphs of Finite Simple Groups

Journal of the London Mathematical Society, 2002

Let G be a finite nonabelian simple group and let Γ be a connected undirected Cayley graph for G. The possible structures for the full automorphism group AutΓ are specified. Then, for certain finite simple groups G, a sufficient condition is given under which G is a normal subgroup of AutΓ. Finally, as an application of these results, several new half-transitive graphs are constructed. Some of these involve the sporadic simple groups G = J 1 , J 4 , Ly and BM, while others fall into two infinite families and involve the Ree simple groups and alternating groups. The two infinite families contain examples of half-transitive graphs of arbitrarily large valency.

A more detailed classification of symmetric cubic graphs

2000

A graph is symmetric if its automorphism group acts transitively on the arcs of , and s-regular if its automorphism group acts regularly on the set of s-arcs of . Tutte (1947, 1959) showed that every cubic finite symmetric cubic graph is s-regular for some s ≤ 5. Djokoviÿc and Miller (1980) proved that there are seven types of arc-transitive

On automorphism groups of Cayley graphs

Periodica Mathematica Hungarica, 1976

It is well-knot~n tha~ its automorphism group A(X o H) must contain the regular subgroup L G corresponding to the set of left multiplication~ by elements of G. This paper is concerned with minimizing the index [A(Xo, t/):L a] for given G, in particular when this index is always greater than 1. If G is a.beli~n but not one of seven exceptional groups, then a Cayley graph of G exists for which this index is at most 2. Nearly complete results for the generalized dicyclic groups are also obtained.

GROUP AUTOMORPHISM STRUCTURES OF SOME GRAPHS

This paper introduces the basic definitions and properties of simple graphs which are mainly covered in [1] and [2]. Each definition and property is supported by examples and diagrams. There are also some basic facts used in this paper which have been demonstrated by other researchers such as [3] and [4]. The main concern and the focus in this paper are on the automorphism groups of some graphs. The final part of this work have been on cubic graphs and the Boolian graph B n. To achieve the main points, the group automorphisms have been applied on the automorphisms of some graphs. The permutation groups played the principle role in the case. This was used to study the nature of the graph automorphisms.

CUBIC SYMMETRIC GRAPHS OF ORDERS 36p AND 36p2

A graph is symmetric, if its automorphism group is transitive on the set of its arcs. In this paper, we classify all the connected cubic symmetric graphs of order 36p and 36p 2 , for each prime p, of which the proof depends on the classification of finite simple groups.

Symmetric cubic graphs of small girth

Journal of Combinatorial Theory, Series B, 2007

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. We show that a symmetric cubic graph of girth at most 9 is either 1-regular or 2 ′-regular (following the notation of Djokovic), or belongs to a small family of exceptional graphs. On the other hand, we show that there are infinitely many 3-regular cubic graphs of girth 10, so that the statement for girth at most 9 cannot be improved to cubic graphs of larger girth. Also we give a characterisation of the 1-or 2 ′-regular cubic graphs of girth g ≤ 9, proving that with five exceptions these are closely related with quotients of the triangle group ∆(2, 3, g) in each case, or of the group x, y | x 2 = y 3 = [x, y] 4 = 1 in the case g = 8. All the 3-transitive cubic graphs and exceptional 1-and 2-regular cubic graphs of girth at most 9 appear in the list of cubic symmetric graphs up to 768 vertices produced by Conder and Dobcsányi (2002); the largest is the 3-regular graph F570 of order 570 (and girth 9). The proofs of the main results are computer-assisted.

CUBIC SYMMETRIC GRAPHS OF ORDER 10p3

Journal of the Korean Mathematical Society, 2013

An automorphism group of a graph is said to be s-regular if it acts regularly on the set of s-arcs in the graph. A graph is s-regular if its full automorphism group is s-regular. In the present paper, all sregular cubic graphs of order 10p 3 are classified for each s ≥ 1 and each prime p.