Let R be a ring and $Z\left(R\right)=Z_l\left(R\right)\cup Z_r\left(R\right)$, where $Z_l\left(R\right)$ and $Z_r\left(R\right)$ are the sets of all left and right zero-divisors of R, respectively. The zero-divisor graph of R, denoted by $\mathrm{\Gamma }\left(R\right)$, is a simple undirected graph with vertex set $Z^{\ast }\left(R\right)=Z\left(R\right)\mathrm{\setminus }\left\{0\right\}$, and two distinct vertices $a,b\in Z^{\ast }\left(R\right)$ are adjacent if and only if ab = 0 or ba = 0. Let $n\ge 2$, ${\mathbb{Z}}_n$ the ring of integers modulo n, and ${\mathrm{M}\mathrm{a}\mathrm{t}}_n\left(q\right)$ the ring of all $n\times n$ matrices over a finite field of q elements. In this article, using the technique on characteristic matrices, we give the value of fixing number of $\mathrm{\Gamma }\left(\prod _{i=1}^n{\mathbb{Z}}_2\right)$. Moreover, we calculate the fixing number and metric dimension of $\mathrm{\Gamma }\left({\mathbb{Z}}_n\right)$ and $\mathrm{\Gamma }\left({\mathrm{M}\mathrm{a}\mathrm{t}}_n\right(q\left)\right)$.

设R是一个环并且$Z\left(\mathrm{电阻}\right)=Z_{升}\left(\mathrm{电阻}\right)\cup Z_r\left(\mathrm{电阻}\right)$， 在哪里 $Z_{升}\left(\mathrm{电阻}\right)$ 和 $Z_r\left(\mathrm{电阻}\right)$分别是R的所有左和右零除数的集合。的零因子图的[R ，记$\mathrm{\Gamma }\left(\mathrm{电阻}\right)$, 是一个带顶点集的简单无向图 $Z^{＊}\left(\mathrm{电阻}\right)=Z\left(\mathrm{电阻}\right)\mathrm{\setminus }\left\{0\right\}$, 和两个不同的顶点 $\mathrm{一种},乙\in Z^{＊}\left(\mathrm{电阻}\right)$相邻当且仅当ab  = 0 或ba  = 0。令$n\ge 2$, ${\mathbb{Z}}_n$整数环模n，和${\mathrm{米}\mathrm{一种}\mathrm{吨}}_n\left(q\right)$ 所有人的戒指 $n\times n$q 个元素的有限域上的矩阵。在本文中，使用特征矩阵上的技术，我们给出了固定数量的值$\mathrm{\Gamma }\left(\prod _{\mathrm{一世}=1}^n{\mathbb{Z}}_2\right)$. 此外，我们计算了固定数和度量维数$\mathrm{\Gamma }\left({\mathbb{Z}}_n\right)$ 和 $\mathrm{\Gamma }\left({\mathrm{米}\mathrm{一种}\mathrm{吨}}_n\right(q\left)\right)$.