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

Abstract

Te use of graph theory is prevalent in the feld of network design, whereby it fnds utility in several domains such as the development of integrated circuits, communication networks, and transportation systems. Te comprehension of partition dimensions may facilitate the enhancement of network designs in terms of efciency and reliability. Let V(G) be a vertex set of a connected graph and S ⊂ V(G), the distance between a vertex v and subset S is defned as d(v, S) = min{d(v, x) | x ∈ S}. An k-ordered partition of V(G) is Rp = {Rp1, Rp2, . . . , Rpk }and the identifcation code of vertex v with respect to Rp is the k-tuple r(v | Rp) = {d(v, Rp1), d(v, Rp2), . . . , d(v, Rpk)}. Te k-partition Rp is said to be a partition resolving if r(v | Rp), ∀v ∈ V(G) are distinct. Partition dimension is the minimum number k in the partition resolving set, symbolized by pd(G). In this paper, we considered the families of graph named as Toeplitz network, and proved that the partition dimension of Tn〈t1, t2〉, where t1 = 2, 3, and gcd (t1, t2) = 1 is constant.