2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/68228In this note we derive some consequences from the Mariño-Vafa formula and the cut-and-join equation, these include unified simple proofs of the $λ_g$ conjecture and some other Hodge integral identities. We also describe a proof of the ELSV formula relating Hurwitz numbers and Hodge integrals by using the cut-and-join equation, following our proof of the Mariño-Vafa formula.16 pages. Final versionAlgebraic GeometryHigh Energy Physics - TheoryMathematical Physicstext