Planar and non-planar wheel-related networks possess edge -graceful labeling

For an integer [Formula: see text], consider a collection of numbers [Formula: see text] and a network [Formula: see text] with [Formula: see text] and [Formula: see text]. The challenge is to create a variety of network designs that allow each edge in the network to employ a single number from the set [Formula: see text] to each edge in the network, ensuring that the amount of accumulation charges at each nodes (connection points) must differ. An edge [Formula: see text]-graceful labeling (E[Formula: see text]GL) of a network [Formula: see text] defined as a bijection [Formula: see text] from the edge set [Formula: see text] to the numbers [Formula: see text] such that the derived transformation [Formula: see text], given by: [Formula: see text], where [Formula: see text], is an injective. Nine designs for wheel-related planar networks and three designs for non-planar networks are presented in this paper. Every one of these networks satisfies our requirements for features and specifications.