Mari\~no-Vafa Formula and Hodge Integral Identities

In this note we derive some consequences from the Mari~no-Vafa formula and the cut-and-join equation, these include unified simple proofs of the ${\lambda }_g$ 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 Mari~no-Vafa formula.