Proof of a conjecture of Graham and Lovász concerning unimodality of coefficients of the distance characteristic polynomial of a tree

Ghodratollah Aalipour; Aida Abiad; Zhanar Berikkyzy; Leslie Hogben; Franklin H.J. Kenter; Jephian C.H. Lin; Michael Tait

doi:10.13001/1081-3810.3493

Proof of a conjecture of Graham and Lovász concerning unimodality of coefficients of the distance characteristic polynomial of a tree

Ghodratollah Aalipour
, Aida Abiad
, Zhanar Berikkyzy
, Leslie Hogben
, Franklin H.J. Kenter
, Jephian C.H. Lin
, Michael Tait

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

4 Citations (Scopus)

Abstract

The conjecture of Graham and Lovász that the (normalized) coefficients of the distance characteristic polynomial of a tree are unimodal is proved; it is also shown that the (normalized) coefficients are log-concave. Upper and lower bounds on the location of the peak are established.

Original language	English
Article number	27
Pages (from-to)	373-380
Number of pages	8
Journal	Electronic Journal of Linear Algebra
Volume	34
DOIs	https://doi.org/10.13001/1081-3810.3493
Publication status	Published - Aug 2018
Externally published	Yes

Funding

Acknowledgment. We thank Ben Braun, Steve Butler, Jay Cummings, Jessica De Silva, Wei Gao, and Kristin Heysse for stimulating discussions, and gratefully acknowledge financial support for this research from NSF 1500662, Elsevier, and the International Linear Algebra Society. We thank the Graduate Research Workshop in Combinatorics (GRWC) where this research took place and the Institute for Mathematics and its Applications (IMA) where connections essential to this research were built.

Keywords

Characteristic polynomial
Distance matrix
Log-concave
Unimodal

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10.13001/1081-3810.3493

Cite this

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abstract = "The conjecture of Graham and Lov{\'a}sz that the (normalized) coefficients of the distance characteristic polynomial of a tree are unimodal is proved; it is also shown that the (normalized) coefficients are log-concave. Upper and lower bounds on the location of the peak are established.",

keywords = "Characteristic polynomial, Distance matrix, Log-concave, Unimodal",

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AU - Abiad, Aida

AU - Berikkyzy, Zhanar

AU - Hogben, Leslie

AU - Kenter, Franklin H.J.

AU - Lin, Jephian C.H.

AU - Tait, Michael

PY - 2018/8

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KW - Distance matrix

KW - Log-concave

KW - Unimodal

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Proof of a conjecture of Graham and Lovász concerning unimodality of coefficients of the distance characteristic polynomial of a tree

Abstract

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Keywords

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