### 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 language | English |
---|---|

Pages (from-to) | 283-288 |

Number of pages | 6 |

Journal | Discrete Applied Mathematics |

Volume | 181 |

DOIs | |

Publication status | Published - Jan 30 2015 |

### Fingerprint

### 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

*Discrete Applied Mathematics*,

*181*, 283-288. https://doi.org/10.1016/j.dam.2014.10.012

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

Research output: Contribution to journal › Article

*Discrete Applied Mathematics*, vol. 181, pp. 283-288. https://doi.org/10.1016/j.dam.2014.10.012

}

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 -