  We compute the expected normalized trace norm (matrix/graph energy) of random
symmetric band circulant matrices and graphs in the limit of large sizes, and
obtain explicit bounds on the rate of convergence to the limit, and on the
probabilities of large deviations. We also show that random symmetric band
Toeplitz matrices have the same limit norm assuming that their band widths
remain small relative to their sizes. We compare the limit norms across a range
of related random matrix and graph ensembles.
