### Abstract

Using the method of seeds and branch duplication, it is shown that for every tree of diameter d <7, there is an Hermitian matrix with as few as d distinct eigenvalues (a known lower bound). For diameter 7, some trees require 8 distinct eigenvalues, but no more; the seeds for which 7 and 8 are the worst case are classified. For trees of diameter d, it is shown that, in general, the minimum number of distinct eigenvalues is bounded by a function of d. Many trees of high diameter permit as few of distinct eigenvalues as the diameter and a conjecture is made that all linear trees are of this type. Several other specific, related observations are made.

Original language | English |
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Pages (from-to) | 557-571 |

Number of pages | 15 |

Journal | Linear & Multilinear Algebra |

Volume | 64 |

Issue number | 3 |

DOIs | |

Publication status | Published - 3 Mar 2016 |

### Keywords

- branch duplication
- diameter
- distinct eigenvalues
- Hermitian matrix
- multiplicities
- tree
- SYMMETRIC-MATRICES
- EIGENVALUES
- GRAPH

## Cite this

Johnson, C. D., & Saiago, C. M. (2016). Diameter minimal trees.

*Linear & Multilinear Algebra*,*64*(3), 557-571. https://doi.org/10.1080/03081087.2015.1057097