24,444 research outputs found
Split digraphs
We generalize the class of split graphs to the directed case and show that
these split digraphs can be identified from their degree sequences. The first
degree sequence characterization is an extension of the concept of splittance
to directed graphs, while the second characterization says a digraph is split
if and only if its degree sequence satisfies one of the Fulkerson inequalities
(which determine when an integer-pair sequence is digraphic) with equality.Comment: 14 pages, 2 figures; Accepted author manuscript (AAM) versio
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
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