آمار
| تعداد نشریات | 6 |
| تعداد شمارهها | 131 |
| تعداد مقالات | 1,634 |
| تعداد مشاهده مقاله | 1,899,064 |
| تعداد دریافت فایل اصل مقاله | 1,758,846 |
Signless Laplacian eigenvalues of the zero divisor graph associated to finite commutative ring $ \mathbb{Z}_{p^{M_{1}}q^{M_{2}}} $ | ||
| Communications in Combinatorics and Optimization | ||
| مقاله 9، دوره 8، شماره 3، آذر 2023، صفحه 561-574 اصل مقاله (434.15 K) | ||
| نوع مقاله: Original paper | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2022.27783.1353 | ||
| نویسندگان | ||
| Shariefuddin Pirzada* 1؛ Bilal Rather2؛ Rezwan Ul Shaban3؛ Tariq Chishti2 | ||
| 1Department of Mathematics, Hazratbal | ||
| 2University of Kashmir | ||
| 3Department of Mathematics, University of Kashmir | ||
| چکیده | ||
| For a commutative ring $R$ with identity $1\neq 0$, let the set $Z(R)$ denote the set of zero-ivisors and let $Z^{*}(R)=Z(R)\setminus \{0\}$ be the set of non-zero zero-divisors of $R$. The zero-divisor graph of $R$, denoted by $\Gamma(R)$, is a simple graph whose vertex set is $Z^{*} (R)$ and two vertices $u, v \in Z^*(R)$ are adjacent if and only if $uv=vu=0$. In this article, we find the signless Laplacian spectrum of the zero divisor graphs $ \Gamma(\mathbb{Z}_{n}) $ for $ n=p^{M_{1}}q^{M_{2}}$, where $ p<q $ are primes and $ M_{1} , M_{2} $ are positive integers. | ||
| کلیدواژهها | ||
| Signless Laplacian matrix؛ zero divisor graph, finite commutative ring, Eulers' s totient function | ||
| مراجع | ||
|
[1] M. Afkhami, Z. Barati, and K. Khashyarmanesh, On the signless Laplacian and normalized Laplacian spectrum of the zero divisor graphs, Ricerche Mat. (2020), https://doi.org/10.1007/s11587–020–00519–3. [2] S. Akbari and A. Mohammadian, On the zero-divisor graph of a commutative ring, J. Algebra 274 (2004), no. 2, 847–855.
[3] D. F. Anderson, T. Asir, A. Badawi, and T. Tamizh Chelvam, Graphs from Rings, Springer, 2021.
[4] D.F. Anderson and P.S. Livingston, The zero divisor graph of a commutative ring, J. Algebra 217 (1999), no. 2, 434–447.
[5] I. Beck, Coloring of a Commutative Rings, J. Algebra 116 (1988), no. 1, 208–226.
[6] 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 Math. 313 (2013), no. 5, 733–741.
[7] S. Chattopadhyay, K.L. Patra, and B.K. Sahoo, Laplacian eigenvalues of the zero divisor graph of the ring $Z_n$, Linear Algebra Appl. 584 (2020), 267–286.
[8] D. Cvetković, P. Rowlison, and S. Simić, An Introduction to the Theory of Graph Spectra, London Math. S. Student Text, 75. Cambridge University Press, Inc. UK, 2010.
[9] H.A. Ganie, B. A Chat, and S. Pirzada, Signless Laplacian energy of a graph and energy of a line graph, Linear Algebra Appl. 544 (2018), 306–324.
[10] T. Koshy, Elementary Number Theory with Applications, Second edition, Academic press, USA, 2002.
[11] P.M. Magi, S.M. Jose, and A. Kishore, Spectrum of the zero-divisor graph on the ring of integers modulo $n$, J. Math. Comput. Sci. 10 (2020), no. 5, 1643–1666.
[12] S. Pirzada, An Introduction to Graph Theory, Universities Press, Orient BlackSwan, Hyderabad, 2012.
[13] S. Pirzada, M. Aijaz, and M.I. Bhat, On zero divisor graphs of the rings $Z_n$, Afrika Matematika 31 (2020), no. 3, 727–737.
[14] S. Pirzada and H.A. Ganie, On the Laplacian eigenvalues of a graph and Laplacian energy, Linear Algebra Appl. 486 (2015), 454–468.
[15] S. Pirzada, B. Rather, R.U. Shaban, and S. Merajuddin, On signless Laplacian spectrum of the zero divisor graphs of the ring $Z_n$, Korean J. Math. 29 (2021), no. 1, 13–24.
[16] B.-F. Wu, Y.-Y. Lou, and C.-X. He, Signless Laplacian and normalized Laplacian on the H-join operation of graphs, Discrete Math. Algorithms Appl. 6 (2014), no. 3, ID: 1450046.
[17] M. Young, Adjacency matrices of zero-divisor graphs of integers modulo n, Involve 8 (2015), no. 5, 753–761. | ||
|
آمار تعداد مشاهده مقاله: 1,037 تعداد دریافت فایل اصل مقاله: 2,009 |
||