We construct some codes, designs and graphs that have the first or second Janko group, J1 or J2, respectively, acting as an automorphism group. We show computationally that the full automorphism group of the design or graph in each case is J1, J2 or J̄2, the extension of J2 by its outer automorphism, and we show that for some of the codes the same is true.

The classical Lie algebras over the complex numbers are all derived from associative algebras by defining a Lie bracket [A,B] = AB−BA, but the five exceptional Lie algebras are often defined directly as Lie algebras. In four cases, however, an alternative approach is available to construct both the algebra and the corresponding Lie group. It is well-known that the Lie group G2 is naturally defi...

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(p—\)(p — \) 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 ...

Motivated by the question of the relative complexities of the Graph Isomorphism and the Graph Automorphism problems we de ne and study the modular graph automor phism problems These are the decision problems modk GA which consist for each k of deciding whether the number of automorphisms of a graph is divisible by k The modk GA problems all turn out to be intermediate in di culty between Graph ...

In this paper, we prove that the full automorphism group of the derangement graph Γn (n ≥ 3) is equal to (R(Sn) ⋊ Inn(Sn)) ⋊ Z2, where R(Sn) and Inn(Sn) are the right regular representation and the inner automorphism group of Sn respectively, and Z2 = 〈φ〉 with the mapping φ : σ φ = σ−1, ∀σ ∈ Sn. Moreover, all orbits on the edge set of Γn (n ≥ 3) are determined.

The problem of finding minimal vertex number of graphs with a given automorphism group is addressed in this article for the case of cyclic groups. This problem was considered earlier by other authors. We give a construction of an undirected graph having 4n+ 6 vertices and automorphism group cyclic of order 4n, n ≥ 1. As a special case we get graphs with 2k + 6 vertices and cyclic automorphism g...

We describe generalized Cayley graphs of rectangular groups, so that we obtain (1) an equivalent condition for two Cayley graphs of a rectangular group to be isomorphic to each other, (2) a necessary and sufficient condition for a generalized Cayley graph of a rectangular group to be (strong) connected, (3) a necessary and sufficient condition for the colour-preserving automorphism group of suc...

It is known that any chordal graph can be uniquely decomposed into simplicial components. Based on this fact, it is shown that for a given chordal graph, its automorphism group can be computed in O((c! · n)O(1)) time, where c denotes the maximum size of simplicial components and n denotes the number of nodes. It is also shown that isomorphism of those chordal graphs can be decided within the sa...

Let φ be an automorphism of a free group Fn of rank n, and let Mφ = Fn oφ Z be the corresponding mapping torus of φ. We study the group Out(Mφ) under certain technical conditions on φ. Moreover, in the case of rank 2, we classify the cases when this group is finite or virtually cyclic, depending on the conjugacy class of the image of φ in GL2(Z). As an application, we solve the isomorphism prob...

We show that if the two parts of a finite bipartite graph have the same degree sequence, then there is a bipartite graph, with the same degree sequences, which is symmetric, in that it has an involutive graph automorphism that interchanges its two parts. To prove this, we study the relationship between symmetric bipartite graphs and graphs with loops.