### Abstract

Motivated by the study of the crossing number of graphs, it is shown that, for trees, the sum of the products of the degrees of the end-vertices of all edges has an upper bound in terms of the sum of all vertex degrees to the power of ϕ^{2}, where ϕ is the golden ratio. The exponent ϕ^{2} is best possible. This inequality is generalized for all graphs with bounded maximum average degree.

Original language | English |
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Pages (from-to) | 742-747 |

Number of pages | 6 |

Journal | American Mathematical Monthly |

Volume | 126 |

Issue number | 8 |

DOIs | |

Publication status | Published - 19 Sep 2019 |

### Cite this

*American Mathematical Monthly*,

*126*(8), 742-747. https://doi.org/10.1080/00029890.2019.1627153

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*American Mathematical Monthly*, vol. 126, no. 8, pp. 742-747. https://doi.org/10.1080/00029890.2019.1627153

**A Golden Ratio Inequality for Vertex Degrees of Graphs.** / Knox, Fiachra; Mohar, Bojan; Wood, David R.

Research output: Contribution to journal › Article › Research › peer-review

TY - JOUR

T1 - A Golden Ratio Inequality for Vertex Degrees of Graphs

AU - Knox, Fiachra

AU - Mohar, Bojan

AU - Wood, David R.

PY - 2019/9/19

Y1 - 2019/9/19

N2 - Motivated by the study of the crossing number of graphs, it is shown that, for trees, the sum of the products of the degrees of the end-vertices of all edges has an upper bound in terms of the sum of all vertex degrees to the power of ϕ2, where ϕ is the golden ratio. The exponent ϕ2 is best possible. This inequality is generalized for all graphs with bounded maximum average degree.

AB - Motivated by the study of the crossing number of graphs, it is shown that, for trees, the sum of the products of the degrees of the end-vertices of all edges has an upper bound in terms of the sum of all vertex degrees to the power of ϕ2, where ϕ is the golden ratio. The exponent ϕ2 is best possible. This inequality is generalized for all graphs with bounded maximum average degree.

UR - http://www.scopus.com/inward/record.url?scp=85073261928&partnerID=8YFLogxK

U2 - 10.1080/00029890.2019.1627153

DO - 10.1080/00029890.2019.1627153

M3 - Article

VL - 126

SP - 742

EP - 747

JO - American Mathematical Monthly

JF - American Mathematical Monthly

SN - 0002-9890

IS - 8

ER -