On the Combinatorial Power of the Weisfeiler-Lehman Algorithm
Authors: Martin Fürer
Abstract: The classical Weisfeiler-Lehman methodology WL[2] makes use of edge colors to provide a strong graph invariant. It is as a minimum as extremely efficient in its means to distinguish non-isomorphic graphs as primarily probably the most excellent algebraic graph invariants. It determines not solely the spectrum of a graph, and the angles between customary basis vectors and the eigenspaces, nevertheless even the angles between projections of regular basis vectors into the eigenspaces. Proper right here, we study the combinatorial vitality of WL[2]. For sufficiently huge okay, WL[k] determines all combinatorial properties of a graph. Many traditionally used combinatorial invariants are determined by WL[k] for small okay. We think about two elementary invariants, the num- ber of cycles Cp of measurement p, and the number of cliques Kp of measurement p. We current that WL[2] determines the number of cycles of lengths as a lot as 6, nevertheless not these of measurement 8. Moreover, WL[2] would not resolve the number of 4-cliques
