  We present an analytical approach for describing spectrally constrained
maximum entropy ensembles of finitely connected regular loopy graphs, valid in
the regime of weak loop-loop interactions. We derive an expression for the
leading two orders of the expected eigenvalue spectrum, through the use of
infinitely many replica indices taking imaginary values. We apply the method to
models in which the spectral constraint reduces to a soft constraint on the
number of triangles, which exhibit `shattering' transitions to phases with
extensively many disconnected cliques, to models with controlled numbers of
triangles and squares, and to models where the spectral constraint reduces to a
count of the number of adjacency matrix eigenvalues in a given interval. Our
predictions are supported by MCMC simulations based on edge swaps with
nontrivial acceptance probabilities.
