101 research outputs found
Two families of graphs that are Cayley on nonisomorphic groups
A number of authors have studied the question of when a graph can be
represented as a Cayley graph on more than one nonisomorphic group. The work to
date has focussed on a few special situations: when the groups are -groups;
when the groups have order ; when the Cayley graphs are normal; or when the
groups are both abelian. In this paper, we construct two infinite families of
graphs, each of which is Cayley on an abelian group and a nonabelian group.
These families include the smallest examples of such graphs that had not
appeared in other results.Comment: 6 page
On the Automorphism Groups of Almost All Circulant Graphs and Digraphs
We attempt to determine the structure of the automorphism group of a generic circulant graph. We first show that almost all circulant graphs have automorphism groups as small as possible. Dobson has conjectured that almost all of the remaining circulant (di)graphs (those whose automorphism groups are not as small as possible) are normal circulant (di)graphs. We show this conjecture is not true in general, but is true if we consider only those circulant (di)graphs whose orders are in a “large” subset of integers. We note that all non-normal circulant (di)graphs can be classified into two natural classes (generalized wreath products, and deleted wreath type), and show that neither of these classes contains almost every non-normal circulant digraph
On the Automorphism Groups of Almost All Circulant Graphs and Digraphs
We attempt to determine the structure of the automorphism group of a generic circulant graph. We first show that almost all circulant graphs have automorphism groups as small as possible. Dobson has conjectured that almost all of the remaining circulant (di)graphs (those whose automorphism groups are not as small as possible) are normal circulant (di)graphs. We show this conjecture is not true in general, but is true if we consider only those circulant (di)graphs whose orders are in a “large” subset of integers. We note that all non-normal circulant (di)graphs can be classified into two natural classes (generalized wreath products, and deleted wreath type), and show that neither of these classes contains almost every non-normal circulant digraph
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