## 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) |
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Pages (from-to) | 1309-1317 |

Number of pages | 9 |

Journal | European Journal of Combinatorics |

Volume | 30 |

Issue number | 5 |

DOIs | |

State | Published - Jul 2009 |

## All Science Journal Classification (ASJC) codes

- Discrete Mathematics and Combinatorics