The Metric Dimension and Partition Dimension of The Amalgamation of Complete Graphs

Abstract

In this paper, there is a section that identifies the aim of the research and makes it possible to suggest exploring the metric dimension and partition dimension of the amalgamation of complete graphs, which we would denote as Amal(K_n, v₀)_t. There are three steps conducted to achieve the research goals in this paper. To begin with, compute the lower bound of the metric dimension and the partition dimension of the graph Amal(K_n, v₀)_t. The second step is to find the upper bounds of the metric dimension and partition dimension of the graph Amal(K_n, v₀)_t by demonstrating that the representation of any vertex in Amal(K_n, v₀)_t is distinct. Finally, the exact values of the metric dimension and partition dimension of the graph Amal(K_n, v₀)_t are found if the lower and upper bounds are determined. The exact value of the metric dimension of the graph Amal(K_n, v₀)_t is denoted as dim(Amal(K_n, v₀)_t), while the exact value of the partition dimension is denoted as pd(Amal(K_n, v₀)_t). In this research, it is found that dim(Amal(K_n, v₀)_t) = (n - 2) for n, t ϵ N with n ≥ 3 and t ≥ 2. It is also found that pd(Amal(K_n, v₀)_t) = n for 2 ≤ t ≤ Cⁿ_n-1.