Twin-free cliques in annihilator graphs of commutative rings

Abstract

For a connected graph [Formula: see text] a clique [Formula: see text] is twin-free if every pair of elements of [Formula: see text] have distinct closed neighborhoods and the number of elements in a twin-free clique of maximum cardinality is called twin-free clique number of [Formula: see text]. The annihilator graph [Formula: see text] of a commutative and unital ring [Formula: see text] is a graph whose vertices are all non-zero zero-divisors of [Formula: see text] and there is an edge between two distinct vertices [Formula: see text] if and only if [Formula: see text] is properly contained in [Formula: see text]. In this paper, twin-free clique number of [Formula: see text] is computed and as an application the strong metric dimension of [Formula: see text] is characterized. Among other things, for a reduced ring [Formula: see text], the forcing strong metric dimension of [Formula: see text] is given.