1,408 research outputs found
A graph theoretic approach to graded identities for matrices
We consider the algebra M_k(C) of k-by-k matrices over the complex numbers
and view it as a crossed product with a group G of order k by embedding G in
the symmetric group S_k via the regular representation and embedding S_k in
M_k(C) in the usual way. This induces a natural G-grading on M_k(C) which we
call a crossed product grading. This grading is the so called elementary
grading defined by any k-tuple (g_1,g_2,..., g_k) of distinct elements g_i in
G. We study the graded polynomial identities for M_k(C) equipped with a crossed
product grading. To each multilinear monomial in the free graded algebra we
associate a directed labeled graph. This approach allows us to give new proofs
of known results of Bahturin and Drensky on the generators of the T-ideal of
identities and the Amitsur-Levitsky Theorem. Our most substantial new result is
the determination of the asymptotic formula for the G-graded codimension of
M_k(C).Comment: 21 pages, 3 figure
The mixing time of the switch Markov chains: a unified approach
Since 1997 a considerable effort has been spent to study the mixing time of
switch Markov chains on the realizations of graphic degree sequences of simple
graphs. Several results were proved on rapidly mixing Markov chains on
unconstrained, bipartite, and directed sequences, using different mechanisms.
The aim of this paper is to unify these approaches. We will illustrate the
strength of the unified method by showing that on any -stable family of
unconstrained/bipartite/directed degree sequences the switch Markov chain is
rapidly mixing. This is a common generalization of every known result that
shows the rapid mixing nature of the switch Markov chain on a region of degree
sequences. Two applications of this general result will be presented. One is an
almost uniform sampler for power-law degree sequences with exponent
. The other one shows that the switch Markov chain on the
degree sequence of an Erd\H{o}s-R\'enyi random graph is asymptotically
almost surely rapidly mixing if is bounded away from 0 and 1 by at least
.Comment: Clarification
On realization graphs of degree sequences
Given the degree sequence of a graph, the realization graph of is the
graph having as its vertices the labeled realizations of , with two vertices
adjacent if one realization may be obtained from the other via an
edge-switching operation. We describe a connection between Cartesian products
in realization graphs and the canonical decomposition of degree sequences
described by R.I. Tyshkevich and others. As applications, we characterize the
degree sequences whose realization graphs are triangle-free graphs or
hypercubes.Comment: 10 pages, 5 figure
On the number of unlabeled vertices in edge-friendly labelings of graphs
Let be a graph with vertex set and edge set , and be a
0-1 labeling of so that the absolute difference in the number of edges
labeled 1 and 0 is no more than one. Call such a labeling
\emph{edge-friendly}. We say an edge-friendly labeling induces a \emph{partial
vertex labeling} if vertices which are incident to more edges labeled 1 than 0,
are labeled 1, and vertices which are incident to more edges labeled 0 than 1,
are labeled 0. Vertices that are incident to an equal number of edges of both
labels we call \emph{unlabeled}. Call a procedure on a labeled graph a
\emph{label switching algorithm} if it consists of pairwise switches of labels.
Given an edge-friendly labeling of , we show a label switching algorithm
producing an edge-friendly relabeling of such that all the vertices are
labeled. We call such a labeling \textit{opinionated}.Comment: 7 pages, accepted to Discrete Mathematics, special issue dedicated to
Combinatorics 201
Hyperoctahedral Eulerian Idempotents, Hodge Decompositions, and Signed Graph Coloring Complexes
Phil Hanlon proved that the coefficients of the chromatic polynomial of a
graph G are equal (up to sign) to the dimensions of the summands in a
Hodge-type decomposition of the top homology of the coloring complex for G. We
prove a type B analogue of this result for chromatic polynomials of signed
graphs using hyperoctahedral Eulerian idempotents
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