The Monitoring Edge-Geodetic Numbers of Some Networks

For a pair [Formula: see text] of vertices and an edge [Formula: see text] of graph [Formula: see text], we say that [Formula: see text] and [Formula: see text] can monitor [Formula: see text] if removing [Formula: see text] would change the distance between [Formula: see text] and [Formula: see text]. A monitoring edge-geodetic (MEG for short) set can monitor all edges of [Formula: see text] and the minimum size of an MEG set is the monitoring edge-geodetic number and is written [Formula: see text]. In this paper, we obtain the MEG numbers of small world networks, hexagonal networks, hex-derived networks, enhanced mesh networks, and hexagonal star networks.