  We analyse the largest eigenvalue of the adjacency matrix of the
configuration model with large degrees, where the latter are treated as hard
constraints. In particular, we compute the expectation of the largest
eigenvalue for degrees that diverge as the number of vertices $n$ tends to
infinity, uniformly on a scale between $1$ and $\sqrt{n}$, and show that a weak
law of large numbers holds. We compare with what was derived in our earlier
paper "Central limit theorem for the principal eigenvalue and eigenvector of
Chung-Lu random graphs" for the Chung-Lu model, which in the regime considered
represents the corresponding configuration model with soft constraints, and
show that the expectation is shifted down by $1$ asymptotically. This shift is
a signature of breaking of ensemble equivalence between the hard and soft (also
known as micro-canonical and canonical) versions of the configuration model.
The latter result generalizes a previous finding in "A spectral signature of
breaking of ensemble equivalence for constrained random graphs" obtained in the
case when all degrees are equal.
