2 research outputs found
Rotational circulant graphs
A Frobenius group is a transitive permutation group which is not regular but
only the identity element can fix two points. Such a group can be expressed as
the semi-direct product of a nilpotent normal subgroup
and another group fixing a point. A first-kind -Frobenius graph is a
connected Cayley graph on with connection set an -orbit on
that generates , where has an even order or is an involution. It is
known that the first-kind Frobenius graphs admit attractive routing and
gossiping algorithms. A complete rotation in a Cayley graph on a group with
connection set is an automorphism of fixing setwise and permuting
the elements of cyclically. It is known that if the fixed-point set of such
a complete rotation is an independent set and not a vertex-cut, then the
gossiping time of the Cayley graph (under a certain model) attains the smallest
possible value. In this paper we classify all first-kind Frobenius circulant
graphs that admit complete rotations, and describe a means to construct them.
This result can be stated as a necessary and sufficient condition for a
first-kind Frobenius circulant to be 2-cell embeddable on a closed orientable
surface as a balanced regular Cayley map. We construct a family of
non-Frobenius circulants admitting complete rotations such that the
corresponding fixed-point sets are independent and not vertex-cuts. We also
give an infinite family of counterexamples to the conjecture that the
fixed-point set of every complete rotation of a Cayley graph is not a
vertex-cut.Comment: Final versio
Frobenius circulant graphs of valency six, Eisenstein-Jacobi networks, and hexagonal meshes
A Frobenius group is a transitive but not regular permutation group such that
only the identity element can fix two points. A finite Frobenius group can be
expressed as with a nilpotent normal subgroup. A
first-kind -Frobenius graph is a Cayley graph on with connection set
an -orbit on generating , where is of even order or consists
of involutions. We classify all 6-valent first-kind Frobenius circulant graphs
such that the underlying kernel is cyclic. We give optimal gossiping and
routing algorithms for such a circulant and compute its forwarding indices,
Wiener indices and minimum gossip time. We also prove that its broadcasting
time is equal to its diameter plus two or three. We prove that all 6-valent
first-kind Frobenius circulants with cyclic kernels are Eisenstein-Jacobi
graphs, the latter being Cayley graphs on quotient rings of the ring of
Eisenstein-Jacobi integers. We also prove that larger Eisenstein-Jacobi graphs
can be constructed from smaller ones as topological covers, and a similar
result holds for 6-valent first-kind Frobenius circulants. As a corollary any
Eisenstein-Jacobi graph with order congruent to 1 modulo 6 and underlying
Eisenstein-Jacobi integer not an associate of a real integer, is a cover of a
6-valent first-kind Frobenius circulant. A distributed real-time computing
architecture known as HARTS or hexagonal mesh is a special 6-valent first-kind
Frobenius circulant.Comment: This is the version to be published in European Journal of
Combinatoric