Metric and strong metric dimension of essential annihilating-ideal graph of commutative rings

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Abstract
The (strong) metric dimension of a connected graph [Formula: see text] is the cardinal number of a (strong) metric basis in [Formula: see text], i.e., the smallest subset of vertices which can determine the position of each vertex by its vector of distances with respect to itself in a unique way (the smallest subset of vertices such that for every pair of vertices [Formula: see text] contains a vertex [Formula: see text] such that either [Formula: see text] or [Formula: see text]. The essential annihilating-ideal graph [Formula: see text] of a commutative unital ring [Formula: see text] is a simple graph whose vertices are nonzero ideals of [Formula: see text] with nonzero annihilator and two distinct vertices [Formula: see text] are joined by an edge if and only if [Formula: see text] is an essential ideal of [Formula: see text]. In this paper, the metric and strong metric dimensions of [Formula: see text] are presented in some explicit formulae, for various classes of rings [Formula: see text].

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