A cubic graph is a generalized structure of a fuzzy graph that gives more precision, flexibility and compatibility to a system when compared with systems that are designed using fuzzy graphs. In this paper, some properties of an edge regular cubic graph are given. Particularly, strongly regular, edge regular and bi-regular cubic graphs are defined and the necessary and sufficient condition for a cubic graph to be strongly regular is studied. Likewise, we have introduced a partially edge regular cubic graph and fully edge regular cubic graph with suitable illustrations. Finally, we gave an application of cubic digraph in travel time.
