Edge Metric Dimension of Some Classes of Toeplitz Networks

Abstract

Toeplitz networks are used as interconnection networks due to their smaller diameter, symmetry, simpler routing, high connectivity, and reliability. The edge metric dimension of a network is recently introduced, and its applications can be seen in several areas including robot navigation, intelligent systems, network designing, and image processing. For a vertex s and an edge g = s 1 s 2 of a connected graph G , the minimum number from distances of s with s 1 and s 2 is called the distance between s and g . If for every two distinct edges s 1 , s 2 ∈ E G , there always exists w 1 ɛ W E ⊆ V G , such that d s 1 , w 1 ≠ d s 2 , w 1 ; then, W E is named as an edge metric generator. The minimum number of vertices in W E is known as the edge metric dimension of G . In this study, we consider four families of Toeplitz networks T n 1,2 , T n 1,3 , T n 1,4 , and T n 1,2,3 and studied their edge metric dimension. We prove that for all n ≥ 4 , e dim T n 1,2 = 4 , for n ≥ 5 , e dim T n 1,3 = 3 , and for n ≥ 6 , e dim T n 1,4 = 3 . We further prove that for all n ≥ 5 , e dim T n 1,2,3 ≤ 6 , and hence, it is bounded.

On the Constant Edge Resolvability of Some Unicyclic and Multicyclic Graphs

“…e edim of various polyphenyl chains was determined by Ahsan et al in [26]. Alrowaili et al computed the edim of very important classes of Toeplitz graphs in [27]. Xing et al compared the edge resolvability with the mixed metric of wheel graphs in [28].…”

Studies of Connected Networks via Fractional Metric Dimension

“…Locating sets have applications in network discovery [6], coin weighting, robot navigation, pattern recognition [7], and joints in networks [8]. Some interesting results of di erent invariants of MD can be seen in [9][10][11][12], and recently, Dalal et al computed edge MD of some Toeplitz networks [13].…”