118 research outputs found
Cycles in the burnt pancake graphs
The pancake graph is the Cayley graph of the symmetric group on
elements generated by prefix reversals. has been shown to have
properties that makes it a useful network scheme for parallel processors. For
example, it is -regular, vertex-transitive, and one can embed cycles in
it of length with . The burnt pancake graph ,
which is the Cayley graph of the group of signed permutations using
prefix reversals as generators, has similar properties. Indeed, is
-regular and vertex-transitive. In this paper, we show that has every
cycle of length with . The proof given is a
constructive one that utilizes the recursive structure of . We also
present a complete characterization of all the -cycles in for , which are the smallest cycles embeddable in , by presenting their
canonical forms as products of the prefix reversal generators.Comment: Added a reference, clarified some definitions, fixed some typos. 42
pages, 9 figures, 20 pages of appendice
Random induced subgraphs of Cayley graphs induced by transpositions
In this paper we study random induced subgraphs of Cayley graphs of the
symmetric group induced by an arbitrary minimal generating set of
transpositions. A random induced subgraph of this Cayley graph is obtained by
selecting permutations with independent probability, . Our main
result is that for any minimal generating set of transpositions, for
probabilities where , a random induced subgraph has a.s. a unique
largest component of size , where
is the survival probability of a specific branching process.Comment: 18 pages, 1 figur
Cayley graphs on the symmetric group generated by initial reversals have unit spectral gap
In a recent paper Gunnells, Scott and Walden have determined the complete
spectrum of the Schreier graph on the symmetric group corresponding to the
Young subgroup and generated by initial reversals. In
particular they find that the first nonzero eigenvalue, or spectral gap, of the
Laplacian is always 1, and report that "empirical evidence" suggests that this
also holds for the corresponding Cayley graph. We provide a simple proof of
this last assertion, based on the decomposition of the Laplacian of Cayley
graphs, into a direct sum of irreducible representation matrices of the
symmetric group.Comment: Shorter version. Published in the Electron. J. of Combinatoric
On the spectral gap of some Cayley graphs on the Weyl group
The Laplacian of a (weighted) Cayley graph on the Weyl group is a
matrix with equal to the order of the group. We show
that for a class of (weighted) generating sets, its spectral gap (lowest
nontrivial eigenvalue), is actually equal to the spectral gap of a matrix associated to a -dimensional permutation representation of
. This result can be viewed as an extension to of an analogous
result valid for the symmetric group, known as `Aldous' spectral gap
conjecture', proven in 2010 by Caputo, Liggett and Richthammer.Comment: Version 1 (v1) contains a mistake. The main result is proved here
under a less general hypothesis than in v1. Main result of v1 is left as a
conjectur
- …