On Laplacian Eigenvalues of the Zero-Divisor Graph Associated to the Ring of Integers Modulo n

Bilal A. Rather; Shariefuddin Pirzada; Tariq A. Naikoo; Yilun Shang

doi:10.3390/math9050482

On Laplacian Eigenvalues of the Zero-Divisor Graph Associated to the Ring of Integers Modulo n

Bilal A. Rather, Shariefuddin Pirzada^*, Tariq A. Naikoo, Yilun Shang^*

^*Corresponding author for this work

Computer Information Sciences Department

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Abstract

Given a commutative ring R with identity 1≠0, let the set Z(R) denote the set of zero-divisors and let Z*(R)=Z(R)∖0 be the set of non-zero zero-divisors of R. The zero-divisor graph of R, denoted by Γ(R), is a simple graph whose vertex set is Z*(R) and each pair of vertices in Z*(R) are adjacent when their product is 0. In this article, we find the structure and Laplacian spectrum of the zero-divisor graphs Γ(Zn) for n=pN1qN2, where p<q are primes and N1,N2 are positive integers.

Original language	English
Article number	482
Number of pages	17
Journal	Mathematics
Volume	9
Issue number	5
DOIs	https://doi.org/10.3390/math9050482
Publication status	Published - 26 Feb 2021

Keywords

Eulers’s totient function
Gaussian integer ring
Integers modulo ring
Laplacian matrix
Zero-divisor graph

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mathematics-09-00482Final published version, 487 KBLicence: CC BY

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title = "On Laplacian Eigenvalues of the Zero-Divisor Graph Associated to the Ring of Integers Modulo n",

abstract = "Given a commutative ring R with identity 1≠0, let the set Z(R) denote the set of zero-divisors and let Z*(R)=Z(R)∖0 be the set of non-zero zero-divisors of R. The zero-divisor graph of R, denoted by Γ(R), is a simple graph whose vertex set is Z*(R) and each pair of vertices in Z*(R) are adjacent when their product is 0. In this article, we find the structure and Laplacian spectrum of the zero-divisor graphs Γ(Zn) for n=pN1qN2, where p<q are primes and N1,N2 are positive integers.",

keywords = "Eulers{\textquoteright}s totient function, Gaussian integer ring, Integers modulo ring, Laplacian matrix, Zero-divisor graph",

author = "Rather, {Bilal A.} and Shariefuddin Pirzada and Naikoo, {Tariq A.} and Yilun Shang",

note = "Funding information: This research received no external funding. The research of S. Pirzada is supported by the SERB-DST research project number MTR/2017/000084.",

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doi = "10.3390/math9050482",

language = "English",

volume = "9",

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T1 - On Laplacian Eigenvalues of the Zero-Divisor Graph Associated to the Ring of Integers Modulo n

AU - Rather, Bilal A.

AU - Pirzada, Shariefuddin

AU - Naikoo, Tariq A.

AU - Shang, Yilun

N1 - Funding information: This research received no external funding. The research of S. Pirzada is supported by the SERB-DST research project number MTR/2017/000084.

PY - 2021/2/26

Y1 - 2021/2/26

N2 - Given a commutative ring R with identity 1≠0, let the set Z(R) denote the set of zero-divisors and let Z*(R)=Z(R)∖0 be the set of non-zero zero-divisors of R. The zero-divisor graph of R, denoted by Γ(R), is a simple graph whose vertex set is Z*(R) and each pair of vertices in Z*(R) are adjacent when their product is 0. In this article, we find the structure and Laplacian spectrum of the zero-divisor graphs Γ(Zn) for n=pN1qN2, where p<q are primes and N1,N2 are positive integers.

AB - Given a commutative ring R with identity 1≠0, let the set Z(R) denote the set of zero-divisors and let Z*(R)=Z(R)∖0 be the set of non-zero zero-divisors of R. The zero-divisor graph of R, denoted by Γ(R), is a simple graph whose vertex set is Z*(R) and each pair of vertices in Z*(R) are adjacent when their product is 0. In this article, we find the structure and Laplacian spectrum of the zero-divisor graphs Γ(Zn) for n=pN1qN2, where p<q are primes and N1,N2 are positive integers.

KW - Eulers’s totient function

KW - Gaussian integer ring

KW - Integers modulo ring

KW - Laplacian matrix

KW - Zero-divisor graph

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DO - 10.3390/math9050482

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VL - 9

JO - Mathematics

JF - Mathematics

IS - 5

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ER -

On Laplacian Eigenvalues of the Zero-Divisor Graph Associated to the Ring of Integers Modulo n

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Keywords

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