### Abstract

We characterize the possible lists of orderedmultiplicities among matrices whose graph is a generalized star (a tree in which atmost one vertex has degree greater than 2) or a double generalized star. Here, the inverse eigenvalue problem (IEP) for symmetric matrices whose graph is a generalized star is settled. The answer is consistent with a conjecture that determination of the possible orderedmultiplicities is equivalent to the IEP for a given tree. Moreover, a key spectral feature of the IEP in the case of generalized stars is shown to characterize them among trees.

Original language | English |
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Pages (from-to) | 311-330 |

Number of pages | 20 |

Journal | Linear Algebra and Its Applications |

Volume | 373 |

Issue number | Suppl. |

DOIs | |

Publication status | Published - 1 Nov 2003 |

Event | Conference on Combinatorial Matrix Theory - Pohang, Korea, Republic of Duration: 14 Jan 2002 → 17 Jan 2002 |

### Keywords

- Hermitian matrices
- Eigenvalues
- Multiplicities
- Trees
- Inverse eigenvalue problems

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## Cite this

Johnson, C. R., Leal-Duarte, A., & Saiago, C. M. (2003). Inverse eigenvalue problems and lists of multiplicities of eigenvalues for matrices whose graph is a tree: the case of generalized stars and double generalized stars.

*Linear Algebra and Its Applications*,*373*(Suppl.), 311-330. https://doi.org/10.1016/S0024-3795(03)00582-2