On Metric Dimension of Nil-Graph of Ideals of Commutative Rings

Abstract

Let [Formula: see text] be a commutative ring with identity and [Formula: see text] be the ideal of all nilpotent elements of [Formula: see text]. Let [Formula: see text] be a nontrivial ideal of [Formula: see text] and there exists a nontrivial ideal [Formula: see text] such that [Formula: see text] The nil-graph of ideals of [Formula: see text] is defined as the graph [Formula: see text] whose vertex set is the set [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. A subset of vertices [Formula: see text] resolves a graph [Formula: see text] and [Formula: see text] is a resolving set of [Formula: see text] if every vertex is uniquely determined by its vector of distances to the vertices of [Formula: see text] In particular, for an ordered subset [Formula: see text] of vertices in a connected graph [Formula: see text] and a vertex [Formula: see text] of [Formula: see text] the metric representation of [Formula: see text] with respect to [Formula: see text] is the [Formula: see text]-vector [Formula: see text] The set [Formula: see text] is a resolving set for [Formula: see text] if [Formula: see text] implies that [Formula: see text] for all pair of vertices, [Formula: see text] A resolving set [Formula: see text] of minimum cardinality is the metric basis for [Formula: see text] and the number of elements in the resolving set of minimum cardinality is the metric dimension of [Formula: see text] If [Formula: see text] for every pair [Formula: see text] of adjacent vertices of [Formula: see text] then [Formula: see text] is called a local metric set of [Formula: see text]. The minimum [Formula: see text] for which [Formula: see text] has a local metric [Formula: see text]-set is the local metric dimension of [Formula: see text] which is denoted by [Formula: see text]. In this paper, we determine metric dimension and local metric dimension of nil-graph of ideals of commutative rings.