Compressed zero-divisor graphs of finite associative rings (Q2191341)

Based on the definition of the zero divisor graph of a ring, introduced the graph of equivalence classes of zero-divisors \(\Gamma_{\sim}(R)\), which was later called compressed zero-divisor graph. The vertex set of \(\Gamma_{\sim}(R)\) is the set of all equivalence classes of non-zero zero-divisors of \(R\), where two zero-divisors are equivalent if and only they have the same annihilator in \(R\). Two equivalence classes \([x]\) and \([y]\) are adjacent in \(\Gamma_{\sim}(R)\) if and only if \(xy = 0\) or \(yx=0\). Thus \(\Gamma_{\sim}(R)\) is obtained from \(\Gamma(R)\) by compressing the vertices of \(\Gamma(R)\) with the same neighbourhood in \(\Gamma(R)\) into one vertex. In the paper under review the authors study the compressed zero-divisor graph of a finite associative ring \(R\). They describe commutative finite rings with compressed zero-divisor graphs of order 2. Moreover, they find all graphs of order 3 that are the compressed zero-divisor graphs of some finite rings.