On realizations of a joint degree matrix

Éva Czabarka, Aaron Dutle, Péter L. Erdos, I. Miklós

Research output: Contribution to journalArticle

16 Citations (Scopus)

Abstract

The joint degree matrix of a graph gives the number of edges between vertices of degree i and degree j for every pair (i,j). One can perform restricted swap operations to transform a graph into another with the same joint degree matrix. We prove that the space of all realizations of a given joint degree matrix over a fixed vertex set is connected via these restricted swap operations. This was claimed before, but there is a flaw in the proof, which we illustrate by example. We also give a simplified proof of the necessary and sufficient conditions for a matrix to be a joint degree matrix, which includes a general method for constructing realizations. Finally, we address the corresponding MCMC methods to sample uniformly from these realizations.

Original languageEnglish
Pages (from-to)283-288
Number of pages6
JournalDiscrete Applied Mathematics
Volume181
DOIs
Publication statusPublished - Jan 30 2015

Fingerprint

Defects
Swap
MCMC Methods
Graph in graph theory
Transform
Necessary Conditions
Sufficient Conditions
Vertex of a graph

Keywords

  • Degree sequence
  • Erdos-Gallai theorem
  • Havel-Hakimi algorithm
  • Joint degree matrix
  • Restricted swap

ASJC Scopus subject areas

  • Applied Mathematics
  • Discrete Mathematics and Combinatorics

Cite this

On realizations of a joint degree matrix. / Czabarka, Éva; Dutle, Aaron; Erdos, Péter L.; Miklós, I.

In: Discrete Applied Mathematics, Vol. 181, 30.01.2015, p. 283-288.

Research output: Contribution to journalArticle

Czabarka, Éva ; Dutle, Aaron ; Erdos, Péter L. ; Miklós, I. / On realizations of a joint degree matrix. In: Discrete Applied Mathematics. 2015 ; Vol. 181. pp. 283-288.
@article{a78a49ac60904aadb24ea59fd6236f10,
title = "On realizations of a joint degree matrix",
abstract = "The joint degree matrix of a graph gives the number of edges between vertices of degree i and degree j for every pair (i,j). One can perform restricted swap operations to transform a graph into another with the same joint degree matrix. We prove that the space of all realizations of a given joint degree matrix over a fixed vertex set is connected via these restricted swap operations. This was claimed before, but there is a flaw in the proof, which we illustrate by example. We also give a simplified proof of the necessary and sufficient conditions for a matrix to be a joint degree matrix, which includes a general method for constructing realizations. Finally, we address the corresponding MCMC methods to sample uniformly from these realizations.",
keywords = "Degree sequence, Erdos-Gallai theorem, Havel-Hakimi algorithm, Joint degree matrix, Restricted swap",
author = "{\'E}va Czabarka and Aaron Dutle and Erdos, {P{\'e}ter L.} and I. Mikl{\'o}s",
year = "2015",
month = "1",
day = "30",
doi = "10.1016/j.dam.2014.10.012",
language = "English",
volume = "181",
pages = "283--288",
journal = "Discrete Applied Mathematics",
issn = "0166-218X",
publisher = "Elsevier",

}

TY - JOUR

T1 - On realizations of a joint degree matrix

AU - Czabarka, Éva

AU - Dutle, Aaron

AU - Erdos, Péter L.

AU - Miklós, I.

PY - 2015/1/30

Y1 - 2015/1/30

N2 - The joint degree matrix of a graph gives the number of edges between vertices of degree i and degree j for every pair (i,j). One can perform restricted swap operations to transform a graph into another with the same joint degree matrix. We prove that the space of all realizations of a given joint degree matrix over a fixed vertex set is connected via these restricted swap operations. This was claimed before, but there is a flaw in the proof, which we illustrate by example. We also give a simplified proof of the necessary and sufficient conditions for a matrix to be a joint degree matrix, which includes a general method for constructing realizations. Finally, we address the corresponding MCMC methods to sample uniformly from these realizations.

AB - The joint degree matrix of a graph gives the number of edges between vertices of degree i and degree j for every pair (i,j). One can perform restricted swap operations to transform a graph into another with the same joint degree matrix. We prove that the space of all realizations of a given joint degree matrix over a fixed vertex set is connected via these restricted swap operations. This was claimed before, but there is a flaw in the proof, which we illustrate by example. We also give a simplified proof of the necessary and sufficient conditions for a matrix to be a joint degree matrix, which includes a general method for constructing realizations. Finally, we address the corresponding MCMC methods to sample uniformly from these realizations.

KW - Degree sequence

KW - Erdos-Gallai theorem

KW - Havel-Hakimi algorithm

KW - Joint degree matrix

KW - Restricted swap

UR - http://www.scopus.com/inward/record.url?scp=84919429939&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84919429939&partnerID=8YFLogxK

U2 - 10.1016/j.dam.2014.10.012

DO - 10.1016/j.dam.2014.10.012

M3 - Article

AN - SCOPUS:84919429939

VL - 181

SP - 283

EP - 288

JO - Discrete Applied Mathematics

JF - Discrete Applied Mathematics

SN - 0166-218X

ER -