본문 바로가기
KERD
논문KCI 등재

Line graphs of unit graphs associated with the direct product of rings

S. Pirzada|Aaqib Altaf
강원경기수학회 (The Kangweon-Kyungki Mathematical Society)|2022.03|한국수학논문집 (Korean Journal of Mathematics)|제30권 제1호|pp. 53-60

1

조회

0

다운로드

0

인용

0

참고문헌

키워드

line graphzero-divisor graphRing of integers modulounit graph
주제분류수학

초록

Let $R$ be a finite commutative ring with non zero identity. The unit graph of $R$ denoted by $G(R)$ is the graph obtained by setting all the elements of $R$ to be the vertices of a graph and two distinct vertices $x$ and $y$ are adjacent if and only if $x+y \in U(R)$, where $U(R)$ denotes the set of units of $R$. In this paper, we find the commutative rings $R$ for which $G(R)$ is a line graph. Also, we find the rings for which the complements of unit graphs are line graphs.

Abstract

Let $R$ be a finite commutative ring with non zero identity. The unit graph of $R$ denoted by $G(R)$ is the graph obtained by setting all the elements of $R$ to be the vertices of a graph and two distinct vertices $x$ and $y$ are adjacent if and only if $x+y \in U(R)$, where $U(R)$ denotes the set of units of $R$. In this paper, we find the commutative rings $R$ for which $G(R)$ is a line graph. Also, we find the rings for which the complements of unit graphs are line graphs.

참고문헌 (23)

  1. M. Afkhami, "Unit graphs of rings of polynomials and power series", Arab. J. Math, 2(3), pp.233~246 (2013)
  2. N. Ashrafi, "Unit graphs associated to rings", Comm. Algabra, 38(8), pp.2851~2871 (2010)
  3. Z. Barati, "Line zero divisor graphs", J. Algebra Appl, 20(9), pp.2150154~ (2021)[DOI]
  4. M. Imran Bhat, "On strong metric dimension of zero-divisor graphs of rings", 한국수학논문집, 27(3), pp.563~580 (2019), 강원경기수학회
  5. Bilal A. Rather, "On Laplacian eigenvalues of the zer divisor graph associated to the ring of integers modulo n", Mathematics, 9(5), pp.482~ (2021)
  6. L. W. Beineke, "Characterizations of derived graphs", J. Comb. Theory, 9, pp.129~135 (1970)
  7. R. P. Grimaldi, "Graphs from rings", Congr. Numer, 71, pp.95~103 (1990)
  8. F. Heydari, "The unit graph of a left Artinian ring", Acta Math. Hungar, 139(1-2), pp.134~146 (2013)
  9. I. Kaplansky, "Commutative Rings" (1974), The University of Chicago Press
  10. S. Kiani, "Classification of rings with unit graphs having domination number less than four", Rend. Sem. Mat. Univ. Padova, 133, pp.173~195 (2015)
  11. H. R. Maimani, "Necessary and sufficient conditions for unit graphs to be Hamiltonian", Pacific J. Math, 249(2), pp.419~429 (2011)
  12. S. Pirzada, "An Introduction to Graph Theory" (2012), Universities press
  13. S. Pirzada, "On distance signless Laplacian spectrum of graphs and spectrum of zero divisor graphs of Zn", Linear and Multilinear Algebra, 70(17), pp.3354~3369 (2020), Informa UK Limited[DOI]
  14. S. Pirzada, "On distance Laplacian spectrum of zero divisor graphs of Zn", Carpathian Math. Publications, 13(1), pp.48~57 (2021)
  15. S. Pirzada, "On signless Laplacian spectrum of the zero divisor graphs of the ring $\mathbb{Z}_{n}$", 한국수학논문집, 29(1), pp.13~24 (2021), 강원경기수학회
  16. S. Pirzada, "Upper dimension and bases of zero divisor graphs of commutative rings", AKCE International J. Graphs Comb, 17(1), pp.168~173 (2020)
  17. S. Pirzada, "Locating sets and numbers of some graphs associated to commutative rings", J. Algebra Appl, 13(7), pp.1450047~ (2014)
  18. S. Pirzada, "On the metric dimension of a zero divisor graph", Comm. Algebra, 45(4), pp.1399~1408 (2017)
  19. Rameez Raja, "On locating numbers and codes of zero divisor graphs associated with commutative rings", J. Algebra Appl, 15(1), pp.1650014~ (2016)[DOI]
  20. H. Su, "Finite commutative rings with higher genus unit graphs", J. Algebra Appl, 14(1), pp.1550002~ (2015)[DOI]
  21. H. Su, "Rings whose unit graphs are planar", Publ. Math. Debrecen, 86(3-4), pp.363~376 (2015)
  22. H. Su, "The diameter of unit graphs of rings", Taiwanese J. Math, 23(1), pp.1~10 (2019)
  23. H. Su, "On the girth of the unit graph of a ring", J. Algebra Appl, 13(2), pp.1350082~ (2014)[DOI]

관련 논문AI추천

A refinement of the unit and unitary Cayley graphs of a finite ringAli Reza Naghipour, Meysam Rezagholibeigi대한수학회 · 대한수학회보2016 · 제53권 제4호Some graph theoretical properties over zero-divisor graphs of special finite commutative ringsM. Togan, N. Akgunes장전수학회 · Advanced Studies in Contemporary Mathematics2012 · 제22권 제2호An extension of annihilating-ideal graph of commutative ringsFatemeh Nabaei, Mahtab Koohi Kerahroodi대한수학회 · 대한수학회논문집2020 · 제35권 제4호Rings whose associated extended zero-divisor graphs are complementedBrahim El Alaoui, Driss Bennis, Raja L’hamri대한수학회 · 대한수학회보2024 · 제61권 제3호On the annihilator graph of group ringsKazem Khashyarmanesh, Mojgan Afkhami, Sepideh Salehifar대한수학회 · 대한수학회보2017 · 제54권 제1호THE IDENTITY-SUMMAND GRAPH OF COMMUTATIVE SEMIRINGSMehdi Khoramdel, Saboura Dolati Pish Hesari, Shahabaddin Ebrahimi Atani대한수학회 · 대한수학회지2014 · 제51권 제1호

이 간행물의 다른 논문 (16건)

Certain results on three-dimensional $f$-Kenmotsu manifolds with conformal Ricci solitonsTarak Mandalpp. 1-10Pseudo - complementation on generalized almost distributive fuzzy latticesYohannes Gedamu Wondifrawpp. 11-23Character analogues of infinite series identities related to generalized non-holomorphic Eisenstein series임성근pp. 25-42Radau quadrature for a rational almost quasi-Hermite-Fej{\'e}r-type interpolationVishnu Narayan Mishra, Shrawan Kumar, Neha Mathur, Pankaj Mathur, Laxmi Rathourpp. 43-51A type of weakly symmetric structure on a Riemannian manifold김재만pp. 61-66
원문 보기