In this thesis, we study the metric dimension of a zero-divisor graph of a ring. For a ring $R$ we define its zero-divisor graph $\Gamma(R)$ as a simple undirected graph whose vertices are zero-divisors and two distinct vertices are adjacent if and only if their product is 0. Metric dimension of such graph is the size of the smallest ordered subset of its vertices for which two distinct vertices in graph have distinct vectors of distances to elements of this subset. Firstly, we study how to limit the metric dimension of a graph, mainly with twin-sets of a graph, subsets of vertices of a graph where vertices are in a same twin-set if they have the same neighbourhoods. Then we closely study the metric dimension of a zero-divisor graph of the ring of integers modulo n and of the ring of matrices over a field.
|
