Utility of the direct product graph for evaluating the L-RWGK. By construction, any given length-L walk w × on the direct product graph G × = G × G with label sequence w ,× (w × ) corresponds to a unique pair of walks {w , w }, with w ∈ W L (G), w ∈ W L (G ), possessing the label sequence w (w ) = w (w ) = w ,× (w × ), and vice versa (giving a bijection). Fig. illustrates. Therefore, all three of the following quantities are equivalent: