There are many proofs of this lemma, which is actually not that difficult to establish; see for instance these previous blog posts for some examples. In this post I would like to record one further proof, based on the spectral decomposition of the adjacency matrix of , which is essentially due to Frieze and Kannan. (Strictly speaking, Frieze and Kannan used a variant of this argument to establish a weaker form of the regularity lemma, but it is not difficult to modify the Frieze-Kannan argument to obtain the usual form of the regularity lemma instead. Some closely related spectral regularity lemmas were also developed by Szegedy.) I found recently (while speaking at the Abel conference in honour of this year’s laureate, Endre Szemerédi) that this particular argument is not as widely known among graph theory experts as I had thought, so I thought I would record it here.