Laplacian spectrum of the unit graph associated to the ring of integers modulo $ pq $

Wafaa Fakieh; Amal Alsaluli; Hanaa Alashwali; Wafaa Fakieh; Amal Alsaluli; Hanaa Alashwali

doi:10.3934/math.2024200

AIMS Mathematics

2024, Volume 9, Issue 2: 4098-4108. doi: 10.3934/math.2024200

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Research article

Laplacian spectrum of the unit graph associated to the ring of integers modulo $ pq $

1.
Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia
2.
Department of Mathematics, Faculty of Science, University of Bisha, Bisha 61922, Saudi Arabia

Received: 10 November 2023 Revised: 15 December 2023 Accepted: 26 December 2023 Published: 12 January 2024
MSC : 05C25, 05C50, 05C76

Let $ R $ be a ring and $ U(R) $ be the set of unit elements of $ R $. The unit graph $ G(R) $ of $ R $ is the graph whose vertices are all the elements of $ R $, defining distinct vertices $ x $ and $ y $ to be adjacent if and only if $ x + y \in U(R) $. The Laplacian spectrum of $ G(\mathbb{Z}_n) $ was studied when $ n = p^{m} $, where $ p $ is a prime and $ m $ is a positive integer. Consequently, in this paper, we study the Laplacian spectrum of $ G(\mathbb{Z}_n) $, for $ n = p_1p_2...p_k $, where $ p_i $ are distinct primes and $ i = 1, 2, ..., k $.
- unit graph,
- Laplacian matrix,
- Laplacian spectrum,
- direct product,
- ring of integers modulo $ n $
Citation: Wafaa Fakieh, Amal Alsaluli, Hanaa Alashwali. Laplacian spectrum of the unit graph associated to the ring of integers modulo $ pq $[J]. AIMS Mathematics, 2024, 9(2): 4098-4108. doi: 10.3934/math.2024200

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Abstract

Let $ R $ be a ring and $ U(R) $ be the set of unit elements of $ R $. The unit graph $ G(R) $ of $ R $ is the graph whose vertices are all the elements of $ R $, defining distinct vertices $ x $ and $ y $ to be adjacent if and only if $ x + y \in U(R) $. The Laplacian spectrum of $ G(\mathbb{Z}_n) $ was studied when $ n = p^{m} $, where $ p $ is a prime and $ m $ is a positive integer. Consequently, in this paper, we study the Laplacian spectrum of $ G(\mathbb{Z}_n) $, for $ n = p_1p_2...p_k $, where $ p_i $ are distinct primes and $ i = 1, 2, ..., k $.

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