On the Spectrum of Generalized Zero-Divisor Graph of the Ring \(\mathbb{Z}_{p^\alpha q^\beta}\)
DOI:
https://doi.org/10.26713/cma.v15i3.2737Keywords:
Zero-divisor graph, Adjacency matrix, EigenvaluesAbstract
The generalized zero-divisor graph of a commutative ring \(R\), denoted by \(\Gamma'(R)\), is a simple (undirected) graph with vertex set \(Z^*(R)\), the set of all nonzero zero-divisors of \(R\) and two distinct vertices \(x\) and \(y\) are adjacent if \(x^ny=0\) or \(y^nx=0\), for some positive integer \(n\). In this paper, we determine the adjacency spectrum of \(\Gamma'(\mathbb{Z}_{p^{\alpha}q^{\beta}})\), where \(p,q\) are distinct primes and \(\alpha, \beta\) are positive integers. Also, we obtain the clique number, stability number, diameter, and the girth of \(\Gamma'(\mathbb{Z}_{p^{\alpha}q^{\beta}})\).
Downloads
References
D. F. Anderson and P. S. Livingston, The zero-divisor graph of a commutative ring, Journal of Algebra 217(2) (1999), 434 – 447, DOI: 10.1006/jabr.1998.7840.
D. F. Anderson, T. Asir, A. Badawi and T. T. Chelvam, Graphs from Rings, 1st edition, Springer, Cham., xvi + 538 pages (2021), DOI: 10.1007/978-3-030-88410-9.
M. F. Atiyah and I. G. MacDonald, Introduction to Commutative Algebra, 1st edition, CRC Press, Boca Raton, 140 pages (1969), DOI: 10.1201/9780429493638.
L. Beaugris, M. Flores, C. Galing, A. Velasquez and E. Tejada, Weak zero-divisor graphs of finite commutative rings, Communications in Mathematics and Applications 15(1) (2024), 1 – 8, DOI: 10.26713/cma.v15i1.2498.
I. Beck, Coloring of commutative rings, Journal of Algebra 116(1) (1988), 208 – 226, DOI: 10.1016/0021-8693(88)90202-5.
D. M. Cardoso, M. A. de Freitas, E. A. Martins and M. Robbiano, Spectra of graphs obtained by a generalization of the join graph operation, Discrete Mathematics 313(5) (2013), 733 – 741, DOI: 10.1016/j.disc.2012.10.016.
C. Godsil and G. Royle, Algebraic Graph Theory, Springer, New York, NY, xix + 443 pages (2001), DOI: 10.1007/978-1-4613-0163-9.
A. Khairnar and B. N. Waphare, Zero-divisor graphs of laurent polynomials and laurent power series, in: Algebra and its Applications, S. Rizvi, A. Ali and V. Filippis (editors), Springer Proceedings in Mathematics & Statistics, Vol. 174, Springer, Singapore, DOI: 10.1007/978-981-10-1651-6_21.
N. Kumbhar, A. Khairnar and B. N. Waphare, Strong zero-divisor graph of rings with involution, Asian-European Journal of Mathematics 16(10) (2023), 2350179, DOI: 10.1142/s1793557123501796.
A. Lande and A. Khairnar, Generalized zero-divisor graph of ∗-rings, arXiv:2403.10161v1 [math.CO], (2024), DOI: 10.48550/arXiv.2403.10161.
P. M. Magi, Sr. M. Jose and A. Kishore, Adjacency matrix and eigenvalues of the zero-divisor graph Γ(Zn), Journal of Mathematical and Computational Science 10(4) (2020), 1285 – 1297, DOI: 10.28919/jmcs/4590.
A. Patil and B. N. Waphare, The zero-divisor graph of a ring with involution, Journal of Algebra and Its Applications 17(03) (2018), 1850050, DOI: 10.1142/S0219498818500500.
S. Pirzada, B. Rather, R. Shaban and T. Chishti, Signless Laplacian eigenvalues of the zero divisor graph associated to finite commutative ring ZpM1 qM2 , Communications in Combinatorics and Optimization 8(3) (2023), 561 – 574, DOI: 10.22049/cco.2022.27783.1353.
S. Pirzada, B. A. Wani and A. Somasundaram, On the eigenvalues of zero-divisor graph associated to finite commutative ring Zpmqn , AKCE International Journal of Graphs and Combinatorics 18(1) (2021), 1 – 6, DOI: 10.1080/09728600.2021.1873060.
S. P. Redmond, The zero-divisor graph of a non-commutative ring, International Journal of Commutative Rings 1(4) (2002), 203 – 211.
M. Young, Adjacency matrices of zero-divisor graphs of integers modulo n, Involve 8(5) (2015), 753 – 761, DOI: 10.2140/involve.2015.8.753.
Downloads
Published
How to Cite
Issue
Section
License
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a CCAL that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work.
