On Certain Bounds for Edge Metric Dimension of Zero-Divisor Graphs Associated with Rings

Given a finite commutative unital ring S having some non-zero elements x, y such that x.y � 0, the elements of S that possess such property are called the zero divisors, denoted byZ(S). We can associate a graph toS with the help of zero-divisor setZ(S), denoted by ζ(S) (called the zero-divisor graph), to study the algebraic properties of the ring S. In this research work, we aim to produce some general bounds for the edge version of metric dimension regarding zero-divisor graphs of S. To do so, we will discuss the zero-divisor graphs for the ring of integers Zm modulo m, some quotient polynomial rings, and the ring of Gaussian integers Zm[i] modulo m. ,en, we prove the general result for the bounds of edge metric dimension of zero-divisor graphs in terms of maximum degree and diameter of ζ(S). In the end, we provide the commutative rings with the same metric dimension, edge metric dimension, and upper dimension.


Introduction
e connection between two mainstream mathematics fields algebra and graph theory was first proposed by Beck [1]. Initially, he introduced the concept of zero-divisor graph associated to a commutative ring S, where he considered every element of a ring S as the vertices of zero-divisor graph and those two distinct vertices r and s are connected for which r · s � 0. Observe that in this case, the 0 vertex is connected to every other vertex. In literature, this type of zero-divisor graph is denoted by ζ°(S). In this work of Beck, his main idea was to present the coloring of a commutative ring. is investigation of coloring of a commutative ring was further analyzed by Anderson and Naseer in [2]. Also, they provided a counter example to Beck who conjectured that clique number cl(S) and chromatic number χ(S) of a ring S are the same by showing that for a finite local ring S, cl(S) � 5 and χ(S) � 6. e zero-divisor graphs by means of zero divisors Z(S) of a unital commutative ring S were studied by Anderson and Livingston in [3], and we will denote this type of zerodivisor graphs by ζ(S). is definition of zero-divisor graph is slightly different from Beck's definition of zero-divisor graph associated to a commutative ring. Observe that, in this case, the element 0 is not considered as the vertex of zerodivisor graph, and so ζ(S) is a subgraph of ζ°(S). Anderson and Livingston presented the interplay between the ring theoretic properties of S and the graph theoretic properties of ζ(S); furthermore, this research provides some fundamental results related to zero-divisor graph ζ(S).
is concept of zero-divisor graphs associated to a unital commutative ring was then extended by means of noncommutative rings by Redmond [4]. He introduced various ways to define the zero-divisor graph associated to a noncommutative ring, which includes both directed and undirected graphs. is work was then continued by Redmond [5] by means of zero-divisor graph of a commutative ring to an ideal-based zero-divisor graph of a commutative ring, where he thought of generalizing this approach by replacing elements whose product is zero with elements whose product lies in some ideal I of S. An ideal-based zerodivisor graph with the vertex set r ∈ S/I: rs ∈ I for some s ∈ S/I { }, where two different vertices r and s are connected if and only if their product lies in the ideal I, is denoted by ζ I (S). Since then, many authors have been working on this and defined the graphs such as unit graphs, zero-divisor graphs of equivalence classes, total graphs, ideal-based zero-divisor graphs, Jacobson graphs, and so on (see, for example, [5][6][7][8][9]). For graph theory, we refer the readers to [10,11], and for basic definitions in ring theory, we refer the readers to [12,13].
Redmond [14] discussed all the commutative rings with unity (up to isomorphism) which produces the zero-divisor graphs on m � 6, 7, . . . , 14 vertices. Moreover, an algorithm is provided to determine all commutative rings (up to isomorphism) with unity. Redmond and Szabo in [15] discussed the zero-divisor graphs of commutative rings with different metrics and upper dimensions. Ali provided a survey on antiregular graphs in [16]. e graph associated to a commutative ring is surprisingly the best demonstration of the properties of the zero-divisor set of a ring. is graph allows and helps us to figure out the algebraic properties of rings using graph theoretic approaches. e authors in [3] discussed some interesting properties of ζ(S). We use the way adopted by Anderson and Livingston in [3], by considering the set of non-zero zero divisors Z(S) as the vertex set for ζ(S).
roughout the paper, S is assumed to be a finite unital commutative ring, unless otherwise stated. Z(S) is the set of non-zero zero divisors as discussed above. A ring S is called local if it has a unique maximal ideal. e annihilator ann(r) of an element r ∈ S is defined as s ∈ S: rs � 0 { }. An element r ∈ S is called nilpotent if r m � 0 for some positive integer m. A ring S is called reduced if it has no non-zero nilpotent elements. Z m � 0, 1, 2, . . . , m − 1 { } is the ring of integers modulo m, Z m [i] � x + i · y: x, y ∈ Z m and i 2 � −1 is the ring of gaussian integers modulo m with respect to the usual complex addition and multiplication, and K m stands for a finite field. e graph associated to the ring of Gaussian integers modulo m was introduced by Osba et al. in [17], and the zero-divisor graph of the ring of gaussian integers Kelenc et al. introduced the concept of edge version of metric dimension in [18]. Consider a graph G having arbitrary vertices r and s and an edge t � rs; the mapping d: { } is the distance between w and t. Any two edges t 1 and t 2 of G are distinguished by a vertex w of G only if d(w, t 1 ) ≠ d(w, t 2 ). Any subset S of vertex set of G is called an edge resolving set for G if every pair of edges of G is distinguished by some vertex set of S. e cardinality of the smallest edge resolving set of G is called the edge metric dimension of G and is denoted by dim E (G). An edge resolving set S of a connected graph G uniquely codes all the edges of G.

Preliminaries
A graph G(V, E) consists of a vertex set V and an edge set E, and the number |V| denotes the order of G, whereas the number |E| denotes the size of G. An edge t ∈ E(G) relates to a pair of distinct vertices, say r and s, written as t � rs. An alternating arrangement among vertices and edges is known as a walk. If we traverse a graph G such that no vertex and edge is repeated, then it is known as a path. If the initial vertex and the terminal vertex in a path are the same, then it is known as a cycle. e distance between two distinct vertices r and s is the number of edges in the smallest path among them, and it is denoted by d(r, s), and if there does not exist a path among them, we define d(r, s) to be infinite. If d(r, s) � 1, then r and s are said to be neighboring vertices.
{ } represents the distance between vertex u and an edge t � rs. e length of the longest path is the diameter of the graph which is denoted by diam(G).
Mathematically, diam(G) � sup d(r, s) { : r and s are distinct vertices in G}.
Any subset H of vertices together with any subset of edges containing those vertices is a subgraph of a graph G; mathematically, we write H ⊂ G. e number of edges in the smallest cycle subgraph in a graph G is called the girth of graph, denoted by gr(G). e maximal complete subgraph of a graph G is called a clique which is denoted by K and |K| � ω(G) is called the clique number. If there is an edge among every pair of vertices in a graph, then it is said to be complete graph which is denoted by K m , where m is the number of vertices. If the vertices of a graph can be partitioned into two disjoint sets, say X and Y such that each vertex of X is adjacent to each vertex in Y, then the graph is said to be complete bipartite graph, and it is usually denoted by K |X|,|Y| or simply K m,n when |X| � m and |Y| � n. A cut vertex is a vertex that when removed from a connected graph creates two or more components of the graph.
Kelenc et al. in [18] discussed the edge metric dimension of the path graph, complete graph, and complete bipartite graph. Since both the metric dimension and the edge metric dimension are closely related, it is feasible to find out graphs for which the metric dimension and the edge metric dimension are the same, as well as for some other graphs G for e edge metric dimension of the path graph P m , cycle graph C m , and the complete graph K m is given in the following results.
Theorem 1 (see [18], Remark 1]). For any integer m ≥ 2, Next, it is shown that for a complete bipartite graph K r,s different from K 1,1 , the edge metric dimension is r + s − 2.

Mathematical Problems in Engineering
Theorem 2 (see [18], Remark 2]). For any complete bipartite graph K r,s such that r ≥ 1 and

Edge Metric Dimension of Graphs Associated with Rings
For a graph G of single vertex, the edge metric dimension is assumed to be zero and for an empty graph, the edge metric dimension is undefined. So, we begin our discussion with the following observation.
Theorem 3. Let S be a finite commutative ring with unity. en, Proof (i) Suppose that dim E (ζ(S)) is finite; then, there exists a minimal edge metric basis for (ζ(S)), So, d(r, e) � 0, 1, 2 or 3 for every r ∈ V(ζ(S)) and e ∈ E(ζ(S)). Hence, |Z(S)| ≤ 4 t , which implies that Z(S) is finite, and hence S is finite. Conversely, given that S is finite, then |Z(S)| is finite, since Z(S) is contained in S. So, dim E (ζ(S)) is finite.
(ii) As we know that edge metric dimension of ζ(S) is undefined whenever S is an integral domain and vice versa, the assertion follows.
e following result gives the edge metric dimension of the zero-divisor graphs of a ring S whenever ζ(S) is isomorphic to P m for some m. □ Proposition 1. Let S be a finite commutative ring with unity. en, dim E (ζ(S)) � 1 if and only if S is isomorphic to one of the following rings: Proof. Suppose that dim E (ζ(S)) � 1; then, by eorem 1, paths are the only graphs whose edge metric dimension is 1, so ζ(S) � P m . Since |Z(S)| is not more than 3 whenever ζ(S) is a path graph by ( [19], Lemma 2.6), ζ(S) is either P 2 or P 3 .
Conversely, the zero-divisor graphs of above given rings are either P 2 or P 3 [14]. Also, the zero-divisor relation is not transitive for these rings. Hence, by eorem 1, dim E (ζ(S)) � 1.
□ Proposition 2. Let S be a finite commutative ring with unity and ζ(S) � C m . en, S is isomorphic to one of these rings: Proof. Given that S is a commutative ring with unity and ζ(S) is a cycle graph, then by ( [3], eorem 2.4) the length of the cycle graph cannot exceed 4. We have shown the zerodivisor graphs of the above given rings in Figure 1. en, dim E (ζ(S)) � 2 if S is isomorphic to one of the following rings: Theorem 4. Let S be a finite commutative ring with unity such that each r ∈ Z(S) is nilpotent. (2) Given that |Z(S)| ≥ 3 and Z(S) 2 ≠ 0 { }, then there exist some r ∈ Z(S) such that r 2 ≠ 0 which implies that there exists s ∈ Z(S) such that d(r, s) ≥ 2. Hence, Z(S)/ t, s { } is an edge metric generator for any vertex t adjacent to r; therefore, dim E (ζ(S)) ≤ |Z(S)| − 2.
By ( [20], Proposition 1), if S is a commutative ring with unity, the cut vertex of ζ(S) is in the center of ζ(S). By ( [20], Lemma 4), it is shown that if S is a finite commutative ring with unity, then ζ(S) has a cut vertex of degree 1 if and only if either there is some r ∈ S such that |ann(r)| � 2 or S is isomorphic to Z 9 or Z 3 [r]/(r 2 ). e next theorem provides the edge metric dimension of ζ(S) when ζ(S) has a cut vertex but not degree 1 vertex. □ Theorem 5. Let S be a finite commutative ring with unity such that |Z(S)| ≥ 3. If ζ(S) has a cut vertex but no degree 1 vertex, then dim E (ζ(S)) � 5.
Proof. Let S be a commutative ring with unity; if ζ(S) has a cut vertex but no degree one vertex, then by ( [20], eorem 3), S is isomorphic to one of the following rings: (1) e zero-divisor graph associated with first four rings is given in Figure 2(a), and for remaining three rings, the zerodivisor graph is given in Figure 2(b). e minimum edge metric generator for graphs in Figures 2(a) and 2(b) is given, respectively, by 4 , v 6 and S 2 � x 1 , x 2 , x 3 , x 4 , x 5 . Hence, in both cases, dim E (ζ(S)) � 5.
We next determine the edge metric dimension of ζ(S), when ζ(S) has exactly one vertex which is adjacent to every other vertex. □ Theorem 6. Let S be a finite commutative ring with unity and S � Z 2 × K for some finite field K.
Proof. First, we suppose that S is a non-local ring and S � Z 2 × K. On the other hand, if S is a local ring and has no cycles, then by ( [20], eorem 2.1), ζ(S) is isomorphic to either P 2 or P 3 , and hence dim E (ζ(S)) � 1. □ Theorem 7. Let S be a finite commutative ring with unity and S � K 1 × K 2 , and both K 1 and K 2 are finite fields with Proof. Given that S � K 1 × K 2 is a finite commutative ring, then each vertex of the form (u, 0) of the zero-divisor graph ζ(S) is adjacent to each (0, v) and vice versa. So, the vertex set of ζ(S) can be partitioned into two disjoint sets, say U � (u, 0): Hence, ζ(S) � K m−1,n−1 and gr(ζ(S)) � 4 which implies by eorem 2 that dim E (ζ(S)) � |K 1 | + |K 2 |-gr(ζ(S)).
We know that we can break down any positive integer m into set of prime numbers, resulting in the original number after multiplying. We are interested in finding the edge metric dimension of ζ(Z m ) when n � 2p and n � pq where p and q are distinct primes. In both cases, the zero-divisor graph is the complete bipartite graph. e zero divisors of Z m when n � pq can be partitioned into two disjoint sets by taking all the multiples of p in one set and the multiples of q in the other set.  Proof. Given that ζ(S) is a graph associated to a commutative ring S and Z(S) is an annihilator ideal, then by definition for any r ∈ Z(S), we have rs � 0 for all s ∈ Z(S), which implies that ζ(S) is a complete graph. Hence, by eorem 1, dim E (ζ(S)) � |Z(S)| − 1.  □ Proposition 5. Let S be a reduced and P 1 , P 2 be two prime ideals such that Proof. First, we aim to prove that Z(S) � P 1 ∪ P 2 . Suppose that r ∈ Z(S)/P 1 ∪ P 2 , so there exists a non-zero s ∈ S such that r.s � 0 ∈ P 1 ∩ P 2 . So, s ∈ P 1 ∩ P 2 , which is a contradiction because P 1 ∩ P 2 � 0 { } as given. Also, P 1 ∩ P 2 ⊆ Z(S), and hence Z(S) � P 1 ∪ P 2 . Now, we claim that ζ(S) is a complete bipartite graph with partite sets Let a, b ∈ V 1 with ab � 0. en, ab ∈ P 2 , and therefore, either a or b ∈ V 2 , a contradiction. us, ζ(S) is a bipartite graph. Now, to show that ζ(S) is a complete bipartite graph, we take a ∈ V 1 and b ∈ V 2 . So, ab ∈ P 1 and ab ∈ P 2 ; since both P 1 and P 2 are ideals, then ab ∈ P 1 ∩ P 2 � 0 { } which implies that ab � 0 and so ζ(S) is a complete bipartite graph. en, ω(ζ(S)) � 2.
Let us now determine the edge metric dimension of the zero-divisor graph of the ring of Gaussian integers ζ(Z m [i]).