In this paper we present a sufficient condition and a necessary condition for Symmetric Nonnegative Inverse Eigenvalue Problem.This condition is independent of the existing realizability criteria.This criterion is recursive, that is, it determines whether a list Λ = {λ 1 , ..., λ n , λ n+1 } is realizable by a nonnegative symmetric matrix, if the list µ = {µ 1 , ..., µ n } associated to Λ is realizable.This result is easy to program and improves some existing criteria.