# On the Planarity of Extended Zero-Divisor Graphs of Local Rings

Authors: Muhammad Ahsan Binyamin; Sheroz Alam Chaudry; Muhammad Salman Abbasi; Abdu Alameri
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

LetR be a commutative ring with unity 1≠0. Recently Bennis et al. defined the concept of extended zero-divisor graph Γ¯R by considering the vertex set VΓ¯R=Z∗R and any two vertices x and y are adjacent if there exist positive integers m and n, such that xmyn=0 with xm≠0 and yn≠0. The main objective of this article is to check the planar property of extended zero-divisor graphs. Also, a complete list of local rings up to order 27 with planar extended zero-divisor graphs has been collected.


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## Body

## 1. Introduction

Ring theory is one of the important branches of algebra dealing with rings and their properties. A ring is a nonempty set together with two binary operations, usually called addition and multiplication, satisfying some specific properties. The rings can be further classified into commutative rings and noncommutative rings. A ring with only one maximal ideal is called a local ring and the order of a local ring is always a prime power number.A graphG consists of two sets called vertex set VG and edge set EG. To draw a graph, we plot vertices as points and join them accordingly to the edge condition. The distance between two vertices x and y is denoted by dx,y and is the length of the shortest path joining them. The diameter of a graph is denoted by diamG and is defined as diamG=maxdx,y|x,y∈VG. The length of the largest cycle in G is called girth of G and is denoted by grG. A path Pn on n vertices is a graph with vertex set v0,v1,…,vn and edge set v0v1,v1v2,v2v3,…,vn−1vn. The friendship graph Fn is obtained by joining n copies of the cycle graph C3 with a common vertex. A graph is complete if all of its vertices are connected by an edge. A complete graph of order n is denoted by Kn. A graph G is bipartite if its vertex set can be partitioned into two set V1 and V2 such that the edges are from the vertex set V1 to vertex set V2. If V1=m and V2=n, then the bipartite graph is denoted by Km,n. A graph is planar if it can be embedded in the plane. The planarity of a graph can be checked by using famous Kuratowski’s theorem which states that a finite graph G is planar if and only if it does not contain a subgraph homeomorphic to K3,3 or K5. For basic definition related to graph theory, the reader can see the book by Chartrand [1].Throughout the paper, all the rings are nonzero commutative rings with identity. Any two elementsx and y of the ring R are called zero-divisors if xy=0. The set of zero-divisors of R is denoted by ZR and Z∗R=Z∖0. Let ΓR denote the zero-divisor graph with the vertex set Z∗R and any two distinct elements x,y∈Z∗R are adjacent if and only if xy=0. The concept of zero-divisor graph was first introduced by Beck [2], while he was interested in the colorings of commutative rings. He considered all the elements of a commutative ring as vertices and any two distinct vertices x and y are adjacent if xy=0. Later Anderson and Livingston [3] modified this definition by changing the vertex set of ΓR by Z∗R and there is an edge between any two vertices x and y if and only if xy=0.Anderson and Livingston [3] first initiated the study of the relation between graph-theoretic properties of zero-divisor graph with the algebraic properties of the considered commutative ring. They proved that ΓR is always a connected graph with diameter ΓR≤3 and grΓR∈2,3,∞. Akbari et al. [4] proved that if R is any local ring and not a field with more than 32 elements, then ΓR is not a planar graph. Belshoff and Chapman [5] extended their result and proved that if R is any local ring and not a field with more than 27 elements, then ΓR is not a planar graph. They also determined all finite local rings R for which ΓR is planar (to read more about the zero-divisor graph, see [3, 6–12]).Recently, Bennis et al. [13] defined the notion of extended zero-divisor graph Γ¯R by considering the vertex set VΓ¯R=Z∗R and any two vertices x and y are adjacent if there exist positive integers m and n, such that xmyn=0 with xm≠0 and yn≠0. It is easy to observe that the graph ΓR is a subgraph of Γ¯R. The extended zero-divisor graph Γ¯R is a connected graph with a diameter of at most 3 and grΓ¯R∈2,3,∞ [13].The main aim of this work is to determine all finite commutative local rings whose extended zero-divisor graphs are planar. We have proved that ifR is a commutative ring, not a field, and R≠16, then Γ¯R is planar if and only if R〈27. Finally, we determined all the commutative local rings of order 16 whose extended zero-divisor graph is planar.


## 2. Results

In this section, we give a complete answer to the planarity of the extended zero-divisor graph of a commutative local ringR. The following results about the planarity of the zero-divisor-graph are useful to prove our results.Proposition 1 (see [5]).
IfR is a finite local ring and R〉27, then ΓR is nonplanar.Proposition 2 (see [5]).
IfR is a finite local ring and R≤27, then ΓR is planar except for the cases when R is isomorphic to F3x,y/x,y2, ℤ9x/3x,x2, F2x,y/x3,xy,y2, F2x,y,z/x,y,z2, ℤ4x,y/x2−2,xy,2x,y2, ℤ4x/2x,x3, ℤ4x,y/2,x,y2, or ℤ8x/2x,x2.
In the next theorem, we give the answer to the planarity of the extended zero-divisor graph.Theorem 1.
LetR be a local ring, neither a field nor R=16, then Γ¯R is planar if and only if R〈27.Proof.
LetΓ¯R be a planar graph, then it does not contain any subdivision of K5 or K3,3. Suppose on contrary that R≥27. From [14], we know that if R=27, then R is isomorphic to F3x,y/x,y2, F3x/x3, ℤ9x/3x,x2, ℤ9x/3x,x2+3, or ℤ27. If R is isomorphic to F3x,y/x,y2 or ℤ9x/3x,x2, then by [5], ΓR is nonplanar and therefore, Γ¯R is also nonplanar. Now if R is isomorphic to ℤ3x/x3, then Z∗R=x,2x,x2,2x2,x+x2,x+2x2,2x+x2,2x+2x2, and Table 1 depicts the edges between the vertices of R. If R is isomorphic to ℤ9x/3x,x2+3, then Z∗R=3,6,x,2x,3+x,3+2x,6+x,6+2x, and Table 2 depicts the edges between the vertices of R. Now if R is isomorphic to ℤ27, then Z∗R=3,6,9,12,15,18,21,24, and Table 3 depicts the edges between the vertices of R.
From Tables1–3, it follow that Γ¯R is isomorphic to a complete graph K8. Since K8 is nonplanar, therefore, Γ¯R is also nonplanar. This gives if R=27, then Γ¯R is nonplanar. Also from [5], we know that if R〉27, then the zero-divisor graph ΓR is nonplanar. Since ΓR is a subgraph of Γ¯R, therefore, Γ¯R is also nonplanar. Thus, if Γ¯R is planar, then R〈27.
SupposeR〈27, then R=4,8,9, or 25. From [14], we know that if R=p2, where p is some prime number, then R is isomorphic to Fpx/xp or ℤp2. Table 4 gives all possible local rings R and their corresponding graphs. Now if R=8, then all possible local rings and their corresponding graphs are given in Table 5. From Tables 4 and 5, it follow that Γ¯R is planar. □Table 1
Vertices and incident edges of the graphΓ¯ℤ3x/x3.
VertexvVertices having an edge with vertexvx2x, x2, 2x2, x+x2, x+2x2, 2x+x2, 2x+2x22xx, x2, 2x2, x+x2, x+2x2, 2x+x2, 2x+2x2x2x, 2x, 2x2, x+x2, x+2x2, 2x+x2, 2x+2x22x2x, 2x, x2, x+x2, x+2x2, 2x+x2, 2x+2x2x+x2x, 2x, x2, 2x2, x+2x2, 2x+x2, 2x+2x2x+2x2x, 2x, x2, 2x2, x+x2, 2x+x2, 2x+2x22x+x2x, 2x, x2, 2x2, x+x2, x+2x2, 2x+2x22x+2x2x, 2x, x2, 2x2, x+x2, x+2x2, 2x+x2Table 2
Vertices and incident edges of the graphΓ¯ℤ9x/3x,x2+3.
VertexvVertices having an edge with vertexv36,x,2x,3+x,3+2x,6+x,6+2x63,x,2x,3+x,3+2x,6+x,6+2xx3,6,2x,3+x,3+2x,6+x,6+2x2x3,6,x,3+x,3+2x,6+x,6+2x3+x3,6,x,2x,3+2x,6+x,6+2x3+2x3,6,x,2x,3+x,6+x,6+2x6+x3,6,x,2x,3+x,3+2x,6+2x6+2x3,6,x,2x,3+x,3+2x,6+xTable 3
Vertices and incident edges of the graphΓ¯ℤ27.
VertexvVertices having an edge with vertexv36,9,12,15,18,21,2463,9,12,15,18,21,2493,6,12,15,18,21,24123,6,9,15,18,21,24153,6,9,12,18,21,24183,6,9,12,15,21,24213,6,9,12,15,18,24243,6,9,12,15,18,21Table 4
Extended zero-divisor graphs corresponding to local rings of orders 4, 9, and 25.
RRZ∗RΓ¯Rℤ2x/x24xK1ℤ442K1ℤ3x/x29x,2xK2ℤ993,6K2ℤ5x/x225x,2x,3x,4xK4ℤ25255,10,15,20K4Table 5
Extended zero-divisor graphs corresponding to local rings of order 8.
RRZ∗RΓ¯RF2x,y/x,y28x,y,x+yK3F2x/x38x,x2,x+x2K3ℤ4x/2x,x282,x,2+xK3ℤ4x/2x,x2−282,x,2+xK3ℤ882,4,6K3Theorem 2.
LetR be a local ring but not a field. If R=16, then Γ¯R is nonplanar except when R is isomorphic to F4x/x2, F2x,y/x2,y2, ℤ4x/x2+x+1, ℤ4x,y/x2,y2,xy−2, or ℤ4x/x2.Proof.
IfR=16, then from [14], we have R is isomorphic to F4x/x2, F2x/x4, F2x,y/x3,y2,xy, F2x,y/x2−y2,xy, F2x,y/x2,y2, F2x,y,z/x,y,z2, ℤ4x/x2+x+1, ℤ4x/x3−2,2x, ℤ4x/x2−2, ℤ4x/x2−2x−2, ℤ4x,y/x2−2,xy,y2,2x, ℤ4x,y/x2−2,xy,y2−2,2x, ℤ4x,y/x2,xy−2,y2, ℤ4x/x3,2x, ℤ4x/x2, ℤ4x/x2−2x, ℤ4x,y/2,x,y2, ℤ8x/2x,x2, ℤ8x/2x,x2−4, or ℤ16. If R is isomorphic to F4x/x2, where F4=0,1,a,a2=a+1 or ℤ4x/x2+x+1, then Z∗R=x,ax,a2x or Z∗R=2,2x,2+2x, respectively. This is easy to see that in both cases Γ¯R≅K3 and therefore, Γ¯R is planar. Now if R is isomorphic to F2x,y/x2,y2 or ℤ4x,y/x2,xy−2,y2 or ℤ4x/x2, then from Tables 6–8, it follow that Γ¯R is isomorphic to friendship graph F3 and therefore, Γ¯R is planar.
From [5], it follows that if R is isomorphic to F2x,y/x3,y2,xy, F2x,y,z/x,y,z2, ℤ4x,y/x2−2,xy,y2,2x, ℤ4x/x3,2x, or ℤ8x/2x,x2, then ΓR is nonplanar. As ΓR is a subgraph of Γ¯R, therefore, Γ¯R is also nonplanar for these cases. This is easy to see that for all the remaining cases Γ¯R≅K7. As K7 is nonplanar, therefore, Γ¯R is also nonplanar.Table 6
Vertices and the incident edges of the graphΓ¯F2x,y/x2,y2.
VertexvVertices having an edge with vertexvxxy, x+xyyxy, y+xyxyx, y, x+y, x+xy, y+xy, x+y+xyx+yxy, x+y+xyx+xyx, xyy+xyy, xyx+y+xyxy, x+yTable 7
Vertices and the incident edges of the graphΓ¯ℤ4x,y/x2,xy−2,y2.
VertexvVertices having an edge with vertexvx2,2+xy2,2+y2x, y, x+y, 2+x, 2+y, 2+x+y2+x2,x2+y2,yx+y2,2+x+yx+y+xy2,x+yTable 8
Vertices and the incident edges of the graphΓ¯ℤ4x/x2.
VertexvVertices having an edge with vertexv22x, 2+2xx2x, 3x2x2,x, 3x, 2+x, 2+2x, 2+3x3xx, 2x2+x2x, 2+3x2+2x2,2x2+3x2x, 2+x


## 3. Conclusion

In this work, we consider the class of extended zero-divisor graphs introduced by Bennis et al. and determine all finite commutative local rings whose extended zero-divisor graph is planar. We have proved that ifR is a commutative ring, not a field, and R≠16, then Γ¯R is planar if and only if R〈27. Finally, all the commutative local rings of order 16 whose extended zero-divisor graph is planar are determined.



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