125 research outputs found
A classification of prime-valent regular Cayley maps on some groups
A Cayley map is a 2-cell embedding of a Cayley graph into an orientable
surface with the same local orientation induced by a cyclic permutation of
generators at each vertex. In this paper, we provide classifications of
prime-valent regular Cayley maps on abelian groups, dihedral groups and
dicyclic groups. Consequently, we show that all prime-valent regular Cayley
maps on dihedral groups are balanced and all prime-valent regular Cayley maps
on abelian groups are either balanced or anti-balanced. Furthermore, we prove
that there is no prime-valent regular Cayley map on any dicyclic group
Regular Cayley maps on dihedral groups with the smallest kernel
Let be a regular Cayley map on the dihedral group
of order and let be the power function associated
with . In this paper it is shown that the kernel Ker of the
power function is a dihedral subgroup of and if then the
kernel Ker is of order at least . Moreover, all are
classified for which Ker is of order . In particular, besides
sporadic maps on and vertices respectively, two infinite families
of non--balanced Cayley maps on are obtained
Regular balanced Cayley maps on
A {\it regular balanced Cayley map} (RBCM for short) on a finite group
is an embedding of a Cayley graph on into a surface, with
some special symmetric property. People have classified RBCM's for cyclic,
dihedral, generalized quaternion, dicyclic, and semi-dihedral groups. In this
paper we classify RBCM's on the group for each prime number
.Comment: 14 pages, to appear on Discrete Mathematic
Quotients of polynomial rings and regular t-balanced Cayley maps on abelian groups
Given a finite group , a regular -balanced Cayley map (RBCM
for short) is a regular Cayley map such that
for all .
In this paper, we clarify a connection between quotients of polynomial rings
and RBCM's on abelian groups, so as to propose a new approach for
classifying RBCM's. We obtain many new results, in particular, a complete
classification for RBCM's on abelian 2-groups.Comment: The previous version of this paper is entitled "A new approach to
regular balanced Cayley maps on abelian -groups
Smooth skew-morphisms of the dihedral groups
A skew-morphism of a finite group is a permutation on such
that and for all
where is an integer function. A
skew-morphism is smooth if for all . The
concept of smooth skew-morphisms is a generalization of that of -balanced
skew-morphisms. The aim of the paper is to develop a general theory of smooth
skew-morphisms. As an application we classify smooth skew-morphisms of the
dihedral groups.Comment: 23page
Complete regular dessins and skew-morphisms of cyclic groups
A dessin is a 2-cell embedding of a connected -coloured bipartite graph
into an orientable closed surface. A dessin is regular if its group of
orientation- and colour-preserving automorphisms acts regularly on the edges.
In this paper we study regular dessins whose underlying graph is a complete
bipartite graph , called -complete regular dessins. The purpose
is to establish a rather surprising correspondence between -complete
regular dessins and pairs of skew-morphisms of cyclic groups. A skew-morphism
of a finite group is a bijection that satisfies the
identity for some function
and fixes the neutral element of~. We show that
every -complete regular dessin determines a pair of
reciprocal skew-morphisms of the cyclic groups and
.
Conversely, can be reconstructed from such a reciprocal pair.
As a consequence, we prove that complete regular dessins, exact bicyclic
groups with a distinguished pair of generators, and pairs of reciprocal
skew-morphisms of cyclic groups are all in one-to-one correspondence. Finally,
we apply the main result to determining all pairs of integers and for
which there exists, up to interchange of colours, exactly one -complete
regular dessin. We show that the latter occurs precisely when every group
expressible as a product of cyclic groups of order and is abelian,
which eventually comes down to the condition
, where is Euler's totient function.Comment: 19papge
Quantum expanders and the quantum entropy difference problem
We define quantum expanders in a natural way. We show that under certain
conditions classical expander constructions generalize to the quantum setting,
and in particular so does the Lubotzky, Philips and Sarnak construction of
Ramanujan expanders from Cayley graphs of the group PGL. We show that this
definition is exactly what is needed for characterizing the complexity of
estimating quantum entropies.Comment: 30 pages, 1 figur
Skew product groups for monolithic groups
Skew morphisms, which generalise automorphisms for groups, provide a
fundamental tool for the study of regular Cayley maps and, more generally, for
finite groups with a complementary factorisation , where is cyclic
and core-free in . In this paper, we classify all examples in which is
monolithic (meaning that it has a unique minimal normal subgroup, and that
subgroup is not abelian) and core-free in . As a consequence, we obtain a
classification of all proper skew morphisms of finite non-abelian simple
groups
Eigenvalues of Cayley graphs
We survey some of the known results on eigenvalues of Cayley graphs and their
applications, together with related results on eigenvalues of Cayley digraphs
and generalizations of Cayley graphs
Quantum expanders and the quantum entropy difference problem
Classical expanders and extractors have numerous applications in computer science. However, it seems these classical objects have no meaningful quantum generalization. This is because it is easy to generate entropy in quantum computation simply by tracing out registers. In this paper we define quantum expanders and extractors in a natural way. We show that this definition is exactly what is needed for showing that QED, the quantum analogue of ED (the entropy difference problem) is QSZK-complete. We also show that quantum expanders exist and with very good parameters in the high min-entropy regime. The first construction is derived from the work of Ambainis and Smith and is based on expander graphs that are based on Cayley graphs of Abelian groups. The drawback of this construction is that it uses logarithmic seed length (yet, this already suffices for showing that QED is QSZK-complete). We also show a quantum analogue of the Lubotzky, Philips and Sarnak construction of Ramanujan expanders from Cayley graphs of PGL. Our construction is a sequence of two steps on the Cayley graph with a basis change in between steps. We believe this quantum analogue of classical Ramanujan expanders is of independent interest
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