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Let T>1 be an integer, and let𝕋=1,2,…,T. We discuss the spectrum of discrete linear second-order eigenvalue problems Δ2ut-1+λmtut=0, t∈𝕋,  u0=uT+1=0, where λ≠0 is a parameter, m:𝕋→ℝ changes sign and mt≠0 on 𝕋. At last, as an application of this spectrum result, we show the existence of sign-changing solutions of discrete nonlinear second-order problems by using bifurcate technique.