Dimensi Matriks Dan Dimensi Partisi Pada Graf Hasil Operasi Korona

Abstract

Let𝐺(𝑉,𝐸)is a connected graph.For an ordered set 𝑊={𝑤1,𝑤2,…,𝑤𝑘} of vertices, 𝑊⊆𝑉(𝐺), and a vertex 𝑣∈𝑉(𝐺), the representation of 𝑣 with respect to 𝑊 is the ordered k-tuple 𝑟(𝑣|𝑊)={𝑑(𝑣,𝑤1),𝑑(𝑣,𝑤2),…,𝑑(𝑣,𝑤𝑘)|∀𝑣∈𝑉(𝐺)}. 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 𝐺. The metric dimension of 𝐺, denoted by 𝑑𝑖𝑚(𝐺), is the number of vertices in a basis of 𝐺. Then, for a subset S of V(G), the distance between u and S is 𝑑(𝑣,𝑆)=𝑚𝑖𝑛{𝑑(𝑣,𝑥)|∀𝑥∈𝑆,∀𝑣∈𝑉(𝐺)}. Let Π=(𝑆1,𝑆2,…,𝑆𝑙)be an ordered l-partition of V(G), for∀𝑆𝑙⊂𝑉(𝐺) dan𝑣∈𝑉(𝐺), the representation of v with respect to Π is the l-vector 𝑟(𝑣|Π)=(𝑑(𝑣,𝑆1),𝑑(𝑣,𝑆2),…,𝑑(𝑣,𝑆𝑙)). The set Π is called a resolving partition for G if the 𝑙−vector 𝑟(𝑣|Π),∀𝑣∈𝑉(𝐺)are distinct. The minimum l for which there is a resolving l-partition of V(G) is the partition dimension of G, denoted by 𝑝𝑑(𝐺). In this paper, we determine the metric dimension and the partition dimension of corona product graphs 𝐾𝑛⨀𝐾𝑛−1, and we get some result that the metric dimension and partition dimension of 𝐾𝑛⨀𝐾𝑛−1respectively is𝑛(𝑛−2) and 2𝑛−1, for𝑛≥3.
Keyword: Metric dimention, partition dimenstion,corona product graphs