On the Planar Property of an Ideal-Based Weakly Zero-Divisor Graph

被引:0
作者
Ghafoor, Asad [1 ]
Zamri, Siti Norziahidayu Amzee [2 ,3 ]
Sarmind, Nor Haniza [4 ]
El-Sanfaz, Mustafa Anis [5 ]
机构
[1] Univ Sultan Zainal Abidin, Fac Informat & Comp, Besut 22000, Terengganu, Malaysia
[2] Univ Sultan Zainal Abidin, Unisza Sci & Med Fdn Ctr, Kuala Nerus 21300, Terengganu, Malaysia
[3] Univ Sultan Zainal Abidin, East Coast Environm Res Inst, Kuala Nerus 21300, Terengganu, Malaysia
[4] Univ Teknol Malaysia, Fac Sci, Dept Math Sci, Johor Baharu 81310, Johor, Malaysia
[5] Qatar Univ, Coll Arts & Sci, Dept Math & Stat, Math Program, Doha 2713, Qatar
来源
MALAYSIAN JOURNAL OF FUNDAMENTAL AND APPLIED SCIENCES | 2024年 / 20卷 / 06期
关键词
Zero-divisor graph; commutative ring; girth; planar graph; graph theory; DIAMETER; GIRTH;
D O I
10.11113/mjfas.v20n6.3597
中图分类号
O [数理科学和化学]; P [天文学、地球科学]; Q [生物科学]; N [自然科学总论];
学科分类号
07 ; 0710 ; 09 ;
摘要
Let R be a commutative ring with a nonzero identity and Z(R) be the set of zero- divisors of R . The weakly zero-divisor graph of R , denoted by WF(R), is the graph with the vertex set Z(R)* = Z(R)\{0}, where two distinct vertices a and b form an edge if ar = bs = rs = 0 for r, s E R\{0}. For an ideal I of R , the ideal-based zero-divisor graph of R , denoted by FI(R), has vertices {a E R\I: ab E I for some b E R\I} and edges {(a, b): ab E I, a, b E R\I, a b}. In this article, an ideal-based weakly zero-divisor graph of R , denoted by WFI(R), is introduced which contains FI(R) as a subgraph and is identical to the graph WF(R) when I = {0}. The relationship between the graphs WFI(R) and WF(R/I) is investigated and the planar property of WFI(R) is studied. The results show that WF(R/I) is isomorphic to a subgraph of WFI(R). For WFI(R) to be planar, some restraints are provided on the size of the ideal I and girth of WFI(R). In conclusion, the results suggest that WFI(R) and WF(R/I) are strongly related and establish necessary and sufficient conditions for WFI(R) to be planar. In addition, rings R with planar WFI(R) are classified.
引用
收藏
页码:1363 / 1374
页数:12
相关论文
共 26 条
[1]   When a zero-divisor graph is planar or a complete r-partite graph [J].
Akbari, S ;
Maimani, HR ;
Yassemi, S .
JOURNAL OF ALGEBRA, 2003, 270 (01) :169-180
[2]   On the diameter and girth of a zero-divisor graph [J].
Anderson, David F. ;
Mulay, S. B. .
JOURNAL OF PURE AND APPLIED ALGEBRA, 2007, 210 (02) :543-550
[3]   GENERALIZATIONS OF THE ZERO-DIVISOR GRAPH [J].
Anderson, David F. ;
McClurkin, Grace .
INTERNATIONAL ELECTRONIC JOURNAL OF ALGEBRA, 2020, 27 :237-262
[4]   BECK COLORING OF A COMMUTATIVE RING [J].
ANDERSON, DD ;
NASEER, M .
JOURNAL OF ALGEBRA, 1993, 159 (02) :500-514
[5]   The zero-divisor graph of a commutative ring [J].
Anderson, DF ;
Livingston, PS .
JOURNAL OF ALGEBRA, 1999, 217 (02) :434-447
[6]   ON THE IDEAL-BASED ZERO-DIVISOR GRAPHS [J].
Ansari-Toroghy, H. ;
Farshadifar, F. ;
Mahboobi-Abkenar, F. .
INTERNATIONAL ELECTRONIC JOURNAL OF ALGEBRA, 2018, 23 :115-130
[7]   On the diameter and girth of ideal-based zero-divisor graphs [J].
Atani, Shahabaddin Ebrahimi ;
Darani, Ahmad Yousefian ;
Puczylowski, Edmund R. .
PUBLICATIONES MATHEMATICAE-DEBRECEN, 2011, 78 (3-4) :607-612
[8]  
Axtell M., 2009, Involve, A Journal of Mathematics, P17
[9]   COLORING OF COMMUTATIVE RINGS [J].
BECK, I .
JOURNAL OF ALGEBRA, 1988, 116 (01) :208-226
[10]   Planar zero-divisor graphs [J].
Belshoff, Richard ;
Chapman, Jeremy .
JOURNAL OF ALGEBRA, 2007, 316 (01) :471-480