Multiset Dimension of Cayley Digraphs of Abelian Groups

Abstract

Given a set W={w₁, w₂,...,w_k} 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 r_m(v|W)={d(v, w₁), d(v, w₂),..., d(v, w_k)}, where d(v, w_i) is the distance from v to w_i for i=1,2,..., k. The set W is an m-resolving set of G if r_m(u|W)≠ r_m(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(C_n(1, t))=t for t ≥ 2 and n ≥ 2t² and md(C_n(1,2,...,t))≤ t for t ≥ 3 and n ≥ 2t. We also prove that the multiset dimension of the Cayley digraph of ℤ_m ⊕ ℤ_n with generating set {(1, 0), (0, 1)} is min{m, n} for m, n ≥ 6.