4,492 research outputs found
Virtually Abelian Quantum Walks
We introduce quantum walks on Cayley graphs of non-Abelian groups. We focus
on the easiest case of virtually Abelian groups, and introduce a technique to
reduce the quantum walk to an equivalent one on an Abelian group with coin
system having larger dimension. We apply the technique in the case of two
quantum walks on virtually Abelian groups with planar Cayley graphs, finding
the exact solution.Comment: 10 pages, 3 figure
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
Automorphisms of Cayley graphs on generalised dicyclic groups
A graph is called a GRR if its automorphism group acts regularly on its
vertex-set. Such a graph is necessarily a Cayley graph. Godsil has shown that
there are only two infinite families of finite groups that do not admit GRRs :
abelian groups and generalised dicyclic groups. Indeed, any Cayley graph on
such a group admits specific additional graph automorphisms that depend only on
the group. Recently, Dobson and the last two authors showed that almost all
Cayley graphs on abelian groups admit no automorphisms other than these obvious
necessary ones. In this paper, we prove the analogous result for Cayley graphs
on the remaining family of exceptional groups: generalised dicyclic groups.Comment: 18 page
Fractional revival on semi-Cayley graphs over abelian groups
In this paper, we investigate the existence of fractional revival on
semi-Cayley graphs over finite abelian groups. We give some necessary and
sufficient conditions for semi-Cayley graphs over finite abelian groups
admitting fractional revival. We also show that integrality is necessary for
some semi-Cayley graphs admitting fractional revival. Moreover, we characterize
the minimum time when semi-Cayley graphs admit fractional revival. As
applications, we give examples of certain Cayley graphs over the generalized
dihedral groups and generalized dicyclic groups admitting fractional revival
Colorings of some Cayley graphs
Cayley graphs are graphs on algebraic structures, typically groups or
group-like structures. In this paper, we have obtained a few results on Cayley
graphs on Cyclic groups, typically powers of cycles, some colorings of powers
of cycles, Cayley graphs on some non-abelian groups, and Cayley graphs on
gyrogroups.Comment: 9 page
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