On the Constant Partition Dimension of Some Generalized Families of Toeplitz Graph

The use of graph theory is prevalent in the field of network design, whereby it finds utility in several domains such as the development of integrated circuits, communication networks, and transportation systems. The comprehension of partition dimensions may facilitate the enhancement of network designs in terms of efficiency and reliability. Let VG be a vertex set of a connected graph and S⊂VG, the distance between a vertex v and subset S is defined as dv,S=mindv,xx∈S. An k-ordered partition of VG is Rp=Rp1,Rp2,…,Rpk and the identification code of vertex v with respect to Rp is the k-tuple rvRp=dv,Rp1,dv,Rp2,…,dv,Rpk. The k-partition Rp is said to be a partition resolving if rvRp, ∀v∈VG are distinct. Partition dimension is the minimum number k in the partition resolving set, symbolized by pdG. In this paper, we considered the families of graph named as Toeplitz network, and proved that the partition dimension of Tnt1,t2, where t1=2,3, and gcdt1,t2=1 is constant.