Abstract
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 by Z S . We can associate a graph to S with the help of zero-divisor set Z 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 ℤ m modulo m , some quotient polynomial rings, and the ring of Gaussian integers ℤ m i modulo m . Then, 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.
Highlights
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
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 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
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)
Summary
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. If there is an edge among every pair of vertices in a graph, it is said to be complete graph which is denoted by Km, 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, the graph is said to be complete bipartite graph, and it is usually denoted by K|X|,|Y| or Km,n when |X| m and |Y| n. Kelenc et al in [18] discussed the edge metric dimension of the path graph, complete graph, and complete bipartite graph. E edge metric dimension of the path graph Pm, cycle graph Cm, and the complete graph Km is given in the following results. For any complete bipartite graph Kr,s such that r ≥ 1 and s ≥ 2, dimE(Kr,s) dim(Kr,s) r + s − 2
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