### Abstract

In 1962, S. L. Hakimi proved necessary and sufficient conditions for a given sequence of positive integers d_{1}, d_{2}, ..., d_{n} to be the degree sequence of a non-separable graph or that of a connected graph. Our goal in this note is to utilize these results to prove closed formulas for the functions d_{n s} (2 m) and d_{c} (2 m), the number of degree sequences with degree sum 2 m representable by non-separable graphs and connected graphs (respectively). Indeed, we give both generating function proofs as well as bijective proofs of the following identities: d_{n s} (2 m) = p (2 m) - p (2 m - 1) - underover(∑, j = 0, m - 2) p (j) and d_{c} (2 m) = p (2 m) - p (m - 1) - 2 underover(∑, j = 0, m - 2) p (j) where p (j) is the number of unrestricted integer partitions of j.

Original language | English (US) |
---|---|

Pages (from-to) | 1309-1317 |

Number of pages | 9 |

Journal | European Journal of Combinatorics |

Volume | 30 |

Issue number | 5 |

DOIs | |

State | Published - Jul 1 2009 |

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### All Science Journal Classification (ASJC) codes

- Computational Theory and Mathematics
- Geometry and Topology
- Theoretical Computer Science

### Cite this

*European Journal of Combinatorics*,

*30*(5), 1309-1317. https://doi.org/10.1016/j.ejc.2008.10.006

}

*European Journal of Combinatorics*, vol. 30, no. 5, pp. 1309-1317. https://doi.org/10.1016/j.ejc.2008.10.006

**Enumeration of the degree sequences of non-separable graphs and connected graphs.** / Rødseth, Øystein J.; Sellers, James Allen; Tverberg, Helge.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Enumeration of the degree sequences of non-separable graphs and connected graphs

AU - Rødseth, Øystein J.

AU - Sellers, James Allen

AU - Tverberg, Helge

PY - 2009/7/1

Y1 - 2009/7/1

N2 - In 1962, S. L. Hakimi proved necessary and sufficient conditions for a given sequence of positive integers d1, d2, ..., dn to be the degree sequence of a non-separable graph or that of a connected graph. Our goal in this note is to utilize these results to prove closed formulas for the functions dn s (2 m) and dc (2 m), the number of degree sequences with degree sum 2 m representable by non-separable graphs and connected graphs (respectively). Indeed, we give both generating function proofs as well as bijective proofs of the following identities: dn s (2 m) = p (2 m) - p (2 m - 1) - underover(∑, j = 0, m - 2) p (j) and dc (2 m) = p (2 m) - p (m - 1) - 2 underover(∑, j = 0, m - 2) p (j) where p (j) is the number of unrestricted integer partitions of j.

AB - In 1962, S. L. Hakimi proved necessary and sufficient conditions for a given sequence of positive integers d1, d2, ..., dn to be the degree sequence of a non-separable graph or that of a connected graph. Our goal in this note is to utilize these results to prove closed formulas for the functions dn s (2 m) and dc (2 m), the number of degree sequences with degree sum 2 m representable by non-separable graphs and connected graphs (respectively). Indeed, we give both generating function proofs as well as bijective proofs of the following identities: dn s (2 m) = p (2 m) - p (2 m - 1) - underover(∑, j = 0, m - 2) p (j) and dc (2 m) = p (2 m) - p (m - 1) - 2 underover(∑, j = 0, m - 2) p (j) where p (j) is the number of unrestricted integer partitions of j.

UR - http://www.scopus.com/inward/record.url?scp=63149117870&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=63149117870&partnerID=8YFLogxK

U2 - 10.1016/j.ejc.2008.10.006

DO - 10.1016/j.ejc.2008.10.006

M3 - Article

AN - SCOPUS:63149117870

VL - 30

SP - 1309

EP - 1317

JO - European Journal of Combinatorics

JF - European Journal of Combinatorics

SN - 0195-6698

IS - 5

ER -