Multiset Dimension of Cayley Digraphs of Abelian Groups

Given a set W={w1, w2,...,wk} of vertices of a connected graph or digraph G, the representation multiset of a vertex v in G with respect to W is the multiset rm(v|W)={d(v, w1), d(v, w2),..., d(v, wk)}, where d(v, wi) is the distance from v to wi for i=1,2,..., k. The set W is an m-resolving set of G if rm(u|W)≠ rm(v|W) for any pair of vertices u and v in G. The minimum cardinality of an m-resolving set of G is called the multiset dimension md(G) of G. In this work, we show that md(Cn(1, t))=t for t ≥ 2 and n ≥ 2t2 and md(Cn(1,2,...,t))≤ t for t ≥ 3 and n ≥ 2t. We also prove that the multiset dimension of the Cayley digraph of ℤm ⊕ ℤn with generating set {(1, 0), (0, 1)} is min{m, n} for m, n ≥ 6.