### Abstract

A vertex v of a graph G is called groupie if the average degree t_{v} of all neighbors of v in G is not smaller than the average degree t_{G} of G. Every graph contains a groupie vertex; the problem of whether or not every simple graph on ≧2 vertices has at least two groupie vertices turned out to be surprisingly difficult. We present various sufficient conditions for a simple graph to contain at least two groupie vertices. Further, we investigate the function f(n) = max min_{v} (t_{v}/t_{G}), where the maximum ranges over all simple graphs on n vertices, and prove that f(n) = 1/42n + o(1). The corresponding result for multigraphs is in sharp contrast with the above. We also characterize trees in which the local average degree t_{v} is constant.

Original language | English |
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Pages (from-to) | 647-661 |

Number of pages | 15 |

Journal | Journal of Graph Theory |

Volume | 18 |

Issue number | 7 |

DOIs | |

Publication status | Published - 1994 |

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### ASJC Scopus subject areas

- Geometry and Topology

### Cite this

*Journal of Graph Theory*,

*18*(7), 647-661. https://doi.org/10.1002/jgt.3190180702