Some graph theoretical properties over zero-divisor graphs of special finite commutative rings

Abstract

Let R be a commutative ring with identity and let ℤ(R) be the set of zero-divisors of R. It has been widely studied the notion of the zero-divisor graph of R which is defined by ΓT(R) = ℤ(R) \{0} such that the 'distinct vertices x and y are adjacent if and only if xy = 0. As main results of this paper, by considering R = ℤ <inf>q×</inf>ℤ <inf>q</inf> for different primes p and q, we prove some graph theoretical properties over Γ(ℤ <inf>p</inf> × ℤ <inf>q</inf>) which are the generalizations of the results in [12].