360 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
Abelian Cayley digraphs with asymptotically large order for any given degree
Abelian Cayley digraphs can be constructed by using a generalization to Z(n) of the concept of congruence in Z. Here we use this approach to present a family of such digraphs, which, for every fixed value of the degree, have asymptotically large number of vertices as the diameter increases. Up to now, the best known large dense results were all non-constructive.Peer ReviewedPostprint (author's final draft
Inverse monoids of partial graph automorphisms
A partial automorphism of a finite graph is an isomorphism between its vertex
induced subgraphs. The set of all partial automorphisms of a given finite graph
forms an inverse monoid under composition (of partial maps). We describe the
algebraic structure of such inverse monoids by the means of the standard tools
of inverse semigroup theory, namely Green's relations and some properties of
the natural partial order, and give a characterization of inverse monoids which
arise as inverse monoids of partial graph automorphisms. We extend our results
to digraphs and edge-colored digraphs as well
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