Finite rings with some restrictions on zero-divisor graphs

The zero-divisor graph $\Gamma(R)$ of an associative ring $R$ is the graph whose vertices are all nonzero zero-divisors (one-sided and two-sided) of $R$, and two distinct vertices $x$ and $y$ are joined by an edge if and only if either $xy=0$ or $yx=0$.
In the present paper, we give full description of finite rings with regular zero-divisor graphs. We also prove some properties of finite rings such that their zero-divisor graphs satisfy the Dirac condition.