Eigenvalues of the perfect matching derangement graph

The perfect matching derangement graph M2n is the graph whose vertex set consists of the perfect matchings of the complete graph K2n such that two vertices (perfect matchings) are adjacent if and only if they have no edges in common, i.e. they are "derangement" with respect to each othe...

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Main Author: Koh, Samuel Zhi Kang
Other Authors: Bernhard Schmidt
Format: Thesis-Doctor of Philosophy
Language:English
Published: Nanyang Technological University 2023
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Online Access:https://hdl.handle.net/10356/168329
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Institution: Nanyang Technological University
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spelling sg-ntu-dr.10356-1683292023-06-01T08:00:48Z Eigenvalues of the perfect matching derangement graph Koh, Samuel Zhi Kang Bernhard Schmidt Ku Cheng Yeaw School of Physical and Mathematical Sciences bernhard@ntu.edu.sg, cyku@ntu.edu.sg Science::Mathematics::Discrete mathematics::Graph theory The perfect matching derangement graph M2n is the graph whose vertex set consists of the perfect matchings of the complete graph K2n such that two vertices (perfect matchings) are adjacent if and only if they have no edges in common, i.e. they are "derangement" with respect to each other. The perfect matching derangement graph is a graph in the association scheme associated with the Gelfand pair (S2n;Hn) where Hn is the hyperoctrahedral group of degree n. It is well-known that each eigenvalue of M2n is associated with a partition. Godsil, Meagher, and Lindzey conjectured that the sign of the eigenvalue of the perfect matching derangement graph M2n is alternating with respect to the number of boxes in the first row of the Young diagram representation of a partition. This is known as the alternating sign property for M2n. In this thesis, we settle the conjecture in the affirmative. Our approach is based on a combinatorial formula for some shifted symmetric functions which allows us to obtain a recurrence formula for the eigenvalues of M2n. Another graph related to M2n is the permutation derangement graph. Our method also provides a new recurrence relation concerning the eigenvalues of this graph. Doctor of Philosophy 2023-05-29T04:46:03Z 2023-05-29T04:46:03Z 2023 Thesis-Doctor of Philosophy Koh, S. Z. K. (2023). Eigenvalues of the perfect matching derangement graph. Doctoral thesis, Nanyang Technological University, Singapore. https://hdl.handle.net/10356/168329 https://hdl.handle.net/10356/168329 10.32657/10356/168329 en This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License (CC BY-NC 4.0). application/pdf Nanyang Technological University
institution Nanyang Technological University
building NTU Library
continent Asia
country Singapore
Singapore
content_provider NTU Library
collection DR-NTU
language English
topic Science::Mathematics::Discrete mathematics::Graph theory
spellingShingle Science::Mathematics::Discrete mathematics::Graph theory
Koh, Samuel Zhi Kang
Eigenvalues of the perfect matching derangement graph
description The perfect matching derangement graph M2n is the graph whose vertex set consists of the perfect matchings of the complete graph K2n such that two vertices (perfect matchings) are adjacent if and only if they have no edges in common, i.e. they are "derangement" with respect to each other. The perfect matching derangement graph is a graph in the association scheme associated with the Gelfand pair (S2n;Hn) where Hn is the hyperoctrahedral group of degree n. It is well-known that each eigenvalue of M2n is associated with a partition. Godsil, Meagher, and Lindzey conjectured that the sign of the eigenvalue of the perfect matching derangement graph M2n is alternating with respect to the number of boxes in the first row of the Young diagram representation of a partition. This is known as the alternating sign property for M2n. In this thesis, we settle the conjecture in the affirmative. Our approach is based on a combinatorial formula for some shifted symmetric functions which allows us to obtain a recurrence formula for the eigenvalues of M2n. Another graph related to M2n is the permutation derangement graph. Our method also provides a new recurrence relation concerning the eigenvalues of this graph.
author2 Bernhard Schmidt
author_facet Bernhard Schmidt
Koh, Samuel Zhi Kang
format Thesis-Doctor of Philosophy
author Koh, Samuel Zhi Kang
author_sort Koh, Samuel Zhi Kang
title Eigenvalues of the perfect matching derangement graph
title_short Eigenvalues of the perfect matching derangement graph
title_full Eigenvalues of the perfect matching derangement graph
title_fullStr Eigenvalues of the perfect matching derangement graph
title_full_unstemmed Eigenvalues of the perfect matching derangement graph
title_sort eigenvalues of the perfect matching derangement graph
publisher Nanyang Technological University
publishDate 2023
url https://hdl.handle.net/10356/168329
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