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 $s_1,s_2\in E\left(G\right)$ , there always exists $w_1\varepsilon W_E\subseteq V\left(G\right)$ , such that $d\left(s_1,w_1\right)\ne d\left(s_2,w_1\right)$ ; 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 $T_n\left(\mathrm{1,2}\right)$ , $T_n\left(\mathrm{1,3}\right)$ , $T_n\left(\mathrm{1,4}\right)$ , and $T_n\left(\mathrm{1,2,3}\right)$ and studied their edge metric dimension. We prove that for all $n\ge 4$ , $e\dim \left(T_n\left(\mathrm{1,2}\right)\right)=4$ , for $n\ge 5$ , $e\dim \left(T_n\left(\mathrm{1,3}\right)\right)=3$ , and for $n\ge 6$ , $e\dim \left(T_n\left(\mathrm{1,4}\right)\right)=3$ . We further prove that for all $n\ge 5$ , $e\dim \left(T_n\left(\mathrm{1,2,3}\right)\right)\le 6$ , and hence, it is bounded.