A k-orbit map is a map with its automorphism group partitioning the set of flags into k orbits. Recently k-orbit maps were studied by Orbanić, Pellicer and Weiss, for k 6 4. In this paper we use symmetry type graphs to extend such study and classify all the types of 5-orbit maps, as well as all self-dual, properly and improperly, symmetry type of k-orbit maps with k 6 7. Moreover, we determine,...

Let   G Aut and  E G   denote the automorphism group and the edge set of a graph  G, respectively. Weinberg’s Theorem states that 4 is a constant sharp upper bound on the ratio     G E G Aut over planar (or spherical) 3-connected graphs  G. We have obtained various analogues of this theorem for nonspherical graphs, introducing two Weinberg-type bounds for an arbitrary closed surfa...

Given two positive integers π and λ, we prove that there exists a finite linear space whose automorphism group has exactly π orbits on points and λ orbits on lines if and only if π ≤ λ.

Denote the n × n toroidal queen’s graph by Qn. We find its automorphism group Aut(Qn) for each positive integer n, showing that for n ≥ 6, Aut(Qn) is generated by the translations, the group of the square, the homotheties, and (for odd n) the automorphism (x, y) → (y + x, y − x). For each n we find the automorphism classes of edges of Qn, in particular showing that for n > 1, Qn is edge-transit...

There are seven binary extremal self-dual doubly-even codes which are known to have a 2-transitive automorphism group. Using representation theoretical methods we show that there are no other such codes, except possibly n = 1 024. We also classify all extremal ternary self-dual and quaternary Hermitian self-dual codes.

Given a connected graph Γ of order n and diameter d, we establish a tight upper bound for the order of the automorphism group of Γ as a function of n and d, and determine the graphs for which the bound is attained.

In this paper we look at the K-theory of a specific C*-algebra closely related to the irrational rotation algebra. Also it is shown that any automorphism of a C*-algebra A induces group automorphisms of K_{1}(A) amd K_{0}(A) in an obvious way. An interesting problem for any C*-algebra A is to find out whether, given an automorphism of K_{0}(A) and an automorphism of K_{1}(A), we can lift them t...

We show that under Martin’s Axiom, the cofinality cf(Aut(Γ)) of the automorphism group of the random graph Γ is 2ω.

Motivated by the question of the relative complexities of the Graph Isomorphism and the Graph Automorphism problems, we deene and study the modular graph automor-phism problems. These are the decision problems mod k-GA which consist, for each k > 1, of deciding whether the number of automorphisms of a graph is divisible by k. The mod k-GA problems all turn out to be intermediate in diiculty bet...

Symmetric graph designs, or SGDs, were deened by Gronau et al. as a common generalisation of symmetric BIBDs and orthogonal double covers. This note gives a classiication of SGDs admitting a 2-transitive automorphism group. There are too many for a complete determination, but in some special cases the determination can be completed, such as those which admit a 3-transitive group, and those with...