745 research outputs found
Towards random uniform sampling of bipartite graphs with given degree sequence
In this paper we consider a simple Markov chain for bipartite graphs with
given degree sequence on vertices. We show that the mixing time of this
Markov chain is bounded above by a polynomial in in case of {\em
semi-regular} degree sequence. The novelty of our approach lays in the
construction of the canonical paths in Sinclair's method.Comment: 47 pages, submitted for publication. In this version we explain
explicitly our main contribution and corrected a serious flaw in the cycle
decompositio
A simple Havel-Hakimi type algorithm to realize graphical degree sequences of directed graphs
One of the simplest ways to decide whether a given finite sequence of
positive integers can arise as the degree sequence of a simple graph is the
greedy algorithm of Havel and Hakimi. This note extends their approach to
directed graphs. It also studies cases of some simple forbidden edge-sets.
Finally, it proves a result which is useful to design an MCMC algorithm to find
random realizations of prescribed directed degree sequences.Comment: 11 pages, 1 figure submitted to "The Electronic Journal of
Combinatorics
Graph realizations constrained by skeleton graphs
In 2008 Amanatidis, Green and Mihail introduced the Joint Degree Matrix (JDM)
model to capture the fundamental difference in assortativity of networks in
nature studied by the physical and life sciences and social networks studied in
the social sciences. In 2014 Czabarka proposed a direct generalization of the
JDM model, the Partition Adjacency Matrix (PAM) model. In the PAM model the
vertices have specified degrees, and the vertex set itself is partitioned into
classes. For each pair of vertex classes the number of edges between the
classes in a graph realization is prescribed. In this paper we apply the new
{\em skeleton graph} model to describe the same information as the PAM model.
Our model is more convenient for handling problems with low number of partition
classes or with special topological restrictions among the classes. We
investigate two particular cases in detail: (i) when there are only two vertex
classes and (ii) when the skeleton graph contains at most one cycle.Comment: 19 page
Studying freeze-out and hadronization in the Landau hydrodynamical model
We study the rapidity spectra in ultra-relativistic heavy ion collisions in
the framework of the Landau hydrodynamical model. We find that thermal smearing
effects improve the agreement with experimental results on pion rapidity
spectra. We describe a simple model of the hadronization and discuss its
consequences regarding the pion multiplicity and the increasing entropy
condition.Comment: 7 pages, 3 figure
On the swap-distances of different realizations of a graphical degree sequence
One of the first graph theoretical problems which got serious attention
(already in the fifties of the last century) was to decide whether a given
integer sequence is equal to the degree sequence of a simple graph (or it is
{\em graphical} for short). One method to solve this problem is the greedy
algorithm of Havel and Hakimi, which is based on the {\em swap} operation.
Another, closely related question is to find a sequence of swap operations to
transform one graphical realization into another one of the same degree
sequence. This latter problem got particular emphases in connection of fast
mixing Markov chain approaches to sample uniformly all possible realizations of
a given degree sequence. (This becomes a matter of interest in connection of --
among others -- the study of large social networks.) Earlier there were only
crude upper bounds on the shortest possible length of such swap sequences
between two realizations. In this paper we develop formulae (Gallai-type
identities) for these {\em swap-distance}s of any two realizations of simple
undirected or directed degree sequences. These identities improves considerably
the known upper bounds on the swap-distances.Comment: to be publishe
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