Edge Metric Dimension of Some Classes of Toeplitz Networks

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=s1s2 of a connected graph G, the minimum number from distances of s with s1 and s2 is called the distance between s and g. If for every two distinct edges s1,s2∈EG, there always exists w1ɛWE⊆VG, such that ds1,w1≠ds2,w1; then, WE is named as an edge metric generator. The minimum number of vertices in WE is known as the edge metric dimension of G. In this study, we consider four families of Toeplitz networks Tn1,2, Tn1,3, Tn1,4, and Tn1,2,3 and studied their edge metric dimension. We prove that for all n≥4, edimTn1,2=4, for n≥5, edimTn1,3=3, and for n≥6, edimTn1,4=3. We further prove that for all n≥5, edimTn1,2,3≤6, and hence, it is bounded.