On the full automorphism group of a graph

While it is easy to characterize the graphs on which a given transitive permutation groupG acts, it is very difficult to characterize the graphsX with Aut (X)=G. We prove here that for the certain transitive permutation groups a simple necessary condition is also sufficient. As a corollary we find that, whenG is ap-group with no homomorphism ontoZ _p wrZ _p , almost all Cayley graphs ofG have automorphism group isomorphic toG.     

On the automorphism group of the Aschbacher graph

A Moore graph is a regular graph of degree k and diameter d with v vertices such that v ≤ 1 + k + k(k ? 1) + ... + k(k ? 1)^d?1. It is known that a Moore graph of degree k ≥ 3 has diameter 2; i.e., it is strongly regular with parameters λ = 0, µ = 1, and v = k ² + 1, where the degree k is equal to 3, 7, or 57. It is unknown whether there exists a Moore graph of degree k = 57. Aschbacher showed that a Moore graph with k = 57 is not a graph of rank 3. In this connection, we call a Moore graph with k = 57 the Aschbacher graph and investigate its automorphism group G without additional assumptions (earlier, it was assumed that G contains an involution).     

On the action of a group on a graph, II

This paper is a continuation of the survey by the author (V.I. Trofimov, On the action of a group on a graph, Acta Appl. Math. 29 (1992) 161–170) on some results concerning groups of automorphisms of locally finite vertex-symmetric graphs.     

The automorphism group of a certain unbounded non-hyperbolic domain

In this paper we determine the automorphism group of the Fock–Bargmann–Hartogs domain _Dn,m$D_{n,m}$ in Cⁿ×C^m${\mathbb{C}}^n\times {\mathbb{C}}^m$ which is defined by the inequality ‖ζ‖²<e^{−μ‖z‖2}${\Vert \zeta \Vert }^2<e^{-\mu {\Vert z\Vert }^2}$.     

Finite structures with prescribed numbers of orbits of their automorphism group

We discuss the problem of existence of finite structures (groups, linear spaces, graphs, ...) with prescribed numbers of orbits of their automorphism groups on the various types of elements. To Dan Hughes for his 80th birthday.     

A nice group structure on the orbit space of unimodular rows

If A is an affine algebra of dimension d ≥ 2, over a perfect field k, where char k ≠ 2 and c.d.₂ k ≤ 1, or if A = R[X], where R is a local, noetherian ring of dimension d ≥ 2, in which 2R = R, then the group structure of W. van der Kallen on the orbit space Um _d+1(A)/E _d+1(A) is given by coordinatewise multiplication via the product formula

On automorphism groups of certain Goppa codes

Goppa codes are linear codes arising from algebraic curves over finite fields. Sufficient conditions are given ensuring that all automorphisms of a Goppa code are inherited from the automorphism group of the curve. In some cases, these conditions are also necessary. The cases of curves with large automorphism groups, notably the Hermitian and the Deligne-Lusztig curves, are investigated in detail. This research was performed within the activity of GNSAGA of the Italian INDAM, with the financial support of the Italian Ministry MIUR, project “Strutture geometriche, combinatorica e loro applicazioni”, PRIN 2006–2007.     

Main eigenvalues and automorphisms of a graph

We study some aspects of the relationship between algebras associated with graphs and automorphism groups. We study an algebra generated by the adjacent matrix of a graph and the all ones matrix, and derive a lower bound for the rank of the automorphism group of a graph. If a graph attains the equality in the above bound, it is calledextremal. We also describe some properties and examples of extremal graphs.     

Main eigenvalues and automorphisms of a graph

We study some aspects of the relationship between algebras associated with graphs and automorphism groups. We study an algebra generated by the adjacent matrix of a graph and the all ones matrix, and derive a lower bound for the rank of the automorphism group of a graph. If a graph attains the equality in the above bound, it is calledextremal. We also describe some properties and examples of extremal graphs.     

On some characterisations of totally unimodular matrices

A characterisation of totally unimodular matrices is derived from a result of Hoffman and Kruskal. It is similar in spirit to a result of Baum and Trotter. Its relation with some other known characterisations is discussed and in the particular case where the matrices have (0, 1) entries, we derive some properties of the associated unimodular hypergraphs. Similar results for balanced and perfect matrices are also reviewed.