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In this paper we deal with circulant and partitioned into $n$-by-$n$circulant blocks matrices and introduce spectral results concerning this classof matrices. The problem of finding lists of complex numbers corresponding to aset of eigenvalues of a nonnegative block matrix with circulant blocks istreated. Along the paper we call realizable list if its elements are theeigenvalues of a nonnegative matrix. The Guo's index $\lambda_0$ of arealizable list is the minimum spectral radius such that the list (up to theinitial spectral radius) together with $\lambda_0$ is realizable. The Guo'sindex of block circulant matrices with circulant blocks is obtained, and inconsequence, necessary and sufficient conditions concerning the NIEP,Nonnegative Inverse Eigenvalue Problem, for the realizability of some spectraare given.

Block matrices and Guo's Index for block circulant matrices with circulant blocks