Dimensi Matriks Dan Dimensi Partisi Pada Graf Hasil Operasi Korona

Abstract

Letðº(ð‘‰,ð¸)is a connected graph.For an ordered set ð‘Š={ð‘¤1,ð‘¤2,…,ð‘¤ð‘˜} of vertices, ð‘ŠâŠ†ð‘‰(ðº), and a vertex ð‘£âˆˆð‘‰(ðº), the representation of ð‘£ with respect to ð‘Š is the ordered k-tuple ð‘Ÿ(ð‘£|ð‘Š)={ð‘‘(ð‘£,ð‘¤1),ð‘‘(ð‘£,ð‘¤2),…,ð‘‘(ð‘£,ð‘¤ð‘˜)|∀ð‘£âˆˆð‘‰(ðº)}. The set W is called a resolving set of G if every vertex of G has a distinct representation. A resolving set containing a minimum number of vertices is called a basis for ðº. The metric dimension of ðº, denoted by ð‘‘ð‘–ð‘š(ðº), is the number of vertices in a basis of ðº. Then, for a subset S of V(G), the distance between u and S is ð‘‘(ð‘£,ð‘†)=ð‘šð‘–ð‘›{ð‘‘(ð‘£,ð‘¥)|∀ð‘¥âˆˆð‘†,∀ð‘£âˆˆð‘‰(ðº)}. Let Π=(ð‘†1,ð‘†2,…,ð‘†ð‘™)be an ordered l-partition of V(G), for∀ð‘†ð‘™âŠ‚ð‘‰(ðº) danð‘£âˆˆð‘‰(ðº), the representation of v with respect to Π is the l-vector ð‘Ÿ(ð‘£|Π)=(ð‘‘(ð‘£,ð‘†1),ð‘‘(ð‘£,ð‘†2),…,ð‘‘(ð‘£,ð‘†ð‘™)). The set Π is called a resolving partition for G if the ð‘™âˆ’vector ð‘Ÿ(ð‘£|Π),∀ð‘£âˆˆð‘‰(ðº)are distinct. The minimum l for which there is a resolving l-partition of V(G) is the partition dimension of G, denoted by ð‘ð‘‘(ðº). In this paper, we determine the metric dimension and the partition dimension of corona product graphs ð¾ð‘›â¨€ð¾ð‘›âˆ’1, and we get some result that the metric dimension and partition dimension of ð¾ð‘›â¨€ð¾ð‘›âˆ’1respectively isð‘›(ð‘›âˆ’2) and 2ð‘›âˆ’1, forð‘›â‰¥3.
Keyword: Metric dimention, partition dimenstion,corona product graphs