On graphs with equal coprime index and clique number

Abstract

Recently, Katre et al. introduced the concept of the coprime index of a graph. They asked to characterize the graphs for which the coprime index is the same as the clique number. In this paper, we partially solve this problem. In fact, we prove that the clique number and the coprime index of a zero-divisor graph of an ordered set and the zero-divisor graph of a ring (Formula presented.) coincide. Also, it is proved that the annihilating ideal graphs, the co-annihilating ideal graphs and the comaximal ideal graphs of commutative rings can be realized as the zero-divisor graphs of specially constructed posets. Hence the coprime index and the clique number coincide for these graphs as well.