On the metric dimension of Buckminsterfullerene-net graph

Abstract

The metric dimension of an arbitrary connected graph <em>G</em>, denoted by dim(<em>G</em>), is the minimum cardinality of the resolving set <em>W</em> of <em>G</em>. An ordered set <em>W = </em>{<em>w<sub>1</sub>, w<sub>2</sub>,..., w<sub>k</sub></em>} is a resolving set of <em>G</em> if for all two different vertices in <em>G</em>, their metric representations are different with respect to <em>W</em>. The metric representation of a vertex <em>v</em> with respect to <em>W</em> is defined as k-tuple <em>r</em>(<em>v</em>|<em>W</em>) = (<em>d</em>(<em>v</em>,<em>w</em><sub>1</sub>), <em>d</em>(<em>v</em>,<em>w</em><sub>2</sub>),..., <em>d</em>(<em>v</em>,<em>w</em><sub>k</sub>)), where <em>d</em>(<em>v</em>,<em>w</em><sub>j</sub>) is the distance between <em>v</em> and <em>w</em><sub>j</sub> for 1 ≤ <em>j</em> ≤ <em>k</em>. The Buckminsterfullerene graph is a 3-reguler graph on 60 vertices containing some cycles <em>C</em><sub>5</sub> and <em>C</em><sub>6</sub>. Let <em>B</em><sub>60</sub><sup>t</sup> denotes the <em>t</em><sup>th</sup>  <em>B</em><sub>60</sub> for 1 ≤ <em>t</em> ≤ <em>m</em> and <em>m</em> ≥ 2. Let <em>v</em><sub>t</sub> be a terminal vertex for each <em>B</em><sub>60</sub><sup>t</sup>. The Buckminsterfullerene-net, denoted by <em>H</em>:<em>=Amal</em>{<em>B</em><sub>60</sub><sup>t</sup>,<em>v</em>| 1 ≤ <em>t</em> ≤ <em>m</em>; <em>m</em> ≥ 2} is a graph constructed from the identification of all terminal vertices <em>v</em><sub>t</sub>, for 1 ≤ <em>t</em> ≤ <em>m</em> and <em>m</em> ≥ 2, into a new vertex, denoted by <em>v</em>. This paper will determine the metric dimension of the Buckminsterfullerene-net graph <em>H</em>.