On the metric dimension of Buckminsterfullerene-net graph

Abstract

The metric dimension of an arbitrary connected graph G, denoted by dim(G), is the minimum cardinality of the resolving set W of G. An ordered set W = {w₁, w₂,..., w_k} is a resolving set of G if for all two different vertices in G, their metric representations are different with respect to W. The metric representation of a vertex v with respect to W is defined as k-tuple r(v|W) = (d(v,w₁), d(v,w₂),..., d(v,w_k)), where d(v,w_j) is the distance between v and w_j for 1 ≤ j ≤ k. The Buckminsterfullerene graph is a 3-reguler graph on 60 vertices containing some cycles C₅ and C₆. Let B₆₀^t denotes the t^th  B₆₀ for 1 ≤ t ≤ m and m ≥ 2. Let v_t be a terminal vertex for each B₆₀^t. The Buckminsterfullerene-net, denoted by H:=Amal{B₆₀^t,v| 1 ≤ t ≤ m; m ≥ 2} is a graph constructed from the identification of all terminal vertices v_t, for 1 ≤ t ≤ m and m ≥ 2, into a new vertex, denoted by v. This paper will determine the metric dimension of the Buckminsterfullerene-net graph H.