A Note on Orderenergetic Graphs and Unrecognized Direct Products

A graph is orderenergetic if its energy is equal to its order. The direct product of graphs offers a simple way of creating new orderenergetic graphs from existing orderenergetic graphs. We show here that two unary graph operations introduced by Joseph in [MATCH Commun. Math. Comput. Chem.87(2022) 703–715] and [MATCH Commun. Math. Comput. Chem. 89(2023) 665–686] for the construction of orderenergetic graphs are in fact unrecognized direct products with two specific graphs. We also briefly discuss an overlooked possibility for constructing new orderenergetic graphs as direct products of non-orderenergetic graphs.