Extending a conjecture of Graham and Lovász on the distance characteristic polynomial

Aida Abiad; Boris Brimkov; Sakander Hayat; Antonina P. Khramova; Jack H. Koolen

doi:10.1016/j.laa.2023.03.027

Extending a conjecture of Graham and Lovász on the distance characteristic polynomial

Aida Abiad (Corresponding author), Boris Brimkov, Sakander Hayat, Antonina P. Khramova, Jack H. Koolen

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Abstract

Graham and Lovász conjectured in 1978 that the sequence of normalized coefficients of the distance characteristic polynomial of a tree of order n is unimodal with the maximum value occurring at [Formula presented]. In this paper we investigate this problem for block graphs. In particular, we prove the unimodality part and we establish the peak for several extremal cases of uniform block graphs with small diameter.

Original language	English
Pages (from-to)	63-82
Number of pages	20
Journal	Linear Algebra and Its Applications
Volume	693
DOIs	https://doi.org/10.1016/j.laa.2023.03.027
Publication status	Published - 15 Jul 2024

Funding

Aida Abiad is partially supported by the FWO (Research Foundation Flanders), grant number 1285921N . Antonina P. Khramova is supported by the NWO (Dutch Science Foundation), grant number OCENW.KLEIN.475 . Jack H. Koolen is partially supported by the National Natural Science Foundation of China (No. 12071454 ), Anhui Initiative in Quantum Information Technologies (No. AHY150000 ) and the National Key R and D Program of China (No. 2020YFA0713100 ).

Keywords

Block graph
Characteristic polynomial
Distance matrix

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abstract = "Graham and Lov{\'a}sz conjectured in 1978 that the sequence of normalized coefficients of the distance characteristic polynomial of a tree of order n is unimodal with the maximum value occurring at [Formula presented]. In this paper we investigate this problem for block graphs. In particular, we prove the unimodality part and we establish the peak for several extremal cases of uniform block graphs with small diameter.",

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AU - Abiad, Aida

AU - Brimkov, Boris

AU - Hayat, Sakander

AU - Khramova, Antonina P.

AU - Koolen, Jack H.

PY - 2024/7/15

Y1 - 2024/7/15

N2 - Graham and Lovász conjectured in 1978 that the sequence of normalized coefficients of the distance characteristic polynomial of a tree of order n is unimodal with the maximum value occurring at [Formula presented]. In this paper we investigate this problem for block graphs. In particular, we prove the unimodality part and we establish the peak for several extremal cases of uniform block graphs with small diameter.

AB - Graham and Lovász conjectured in 1978 that the sequence of normalized coefficients of the distance characteristic polynomial of a tree of order n is unimodal with the maximum value occurring at [Formula presented]. In this paper we investigate this problem for block graphs. In particular, we prove the unimodality part and we establish the peak for several extremal cases of uniform block graphs with small diameter.

KW - Block graph

KW - Characteristic polynomial

KW - Distance matrix

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DO - 10.1016/j.laa.2023.03.027

M3 - Article

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JF - Linear Algebra and Its Applications

ER -

Extending a conjecture of Graham and Lovász on the distance characteristic polynomial

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Keywords

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