Fixing number and metric dimension of a zero-divisor graph associated with a ring
Shikun Ou, Dein Wong, Fenglei Tian, Qi Zhou · Linear and Multilinear Algebra · 2020
Let R be a ring and Z(R)=Zl(R)∪Zr(R), where Zl(R) and Zr(R) are the sets of all left and right zero-divisors of R, respectively. The zero-divisor graph of R, denoted by Γ(R), is a simple undirected graph with vertex set Z∗(R)=Z(R)∖{0}, and two distinct vertices a,b∈Z∗(R) are adjacent if and only if ab = 0 or ba = 0. Let n≥2, Zn the ring of integers modulo n, and Matn(q) the ring of all n×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 Γ(∏i=1nZ2). Moreover, we calculate the fixing number and metric dimension of Γ(Zn) and Γ(Matn(q)).