### 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

**Local and global average degree in graphs and multigraphs.** / Bertram, E.; Erds, P.; Horák, P.; Širáň, J.; Tuza, Z.

Research output: Contribution to journal › Article

*Journal of Graph Theory*, vol. 18, no. 7, pp. 647-661. https://doi.org/10.1002/jgt.3190180702

}

TY - JOUR

T1 - Local and global average degree in graphs and multigraphs

AU - Bertram, E.

AU - Erds, P.

AU - Horák, P.

AU - Širáň, J.

AU - Tuza, Z.

PY - 1994

Y1 - 1994

N2 - A vertex v of a graph G is called groupie if the average degree tv of all neighbors of v in G is not smaller than the average degree tG 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 minv (tv/tG), 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 tv is constant.

AB - A vertex v of a graph G is called groupie if the average degree tv of all neighbors of v in G is not smaller than the average degree tG 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 minv (tv/tG), 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 tv is constant.

UR - http://www.scopus.com/inward/record.url?scp=84987486497&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84987486497&partnerID=8YFLogxK

U2 - 10.1002/jgt.3190180702

DO - 10.1002/jgt.3190180702

M3 - Article

VL - 18

SP - 647

EP - 661

JO - Journal of Graph Theory

JF - Journal of Graph Theory

SN - 0364-9024

IS - 7

ER -