  A Chung-Lu random graph is an inhomogeneous Erd\H{o}s-R\'enyi random graph in
which vertices are assigned average degrees, and pairs of vertices are
connected by an edge with a probability that is proportional to the product of
their average degrees, independently for different edges. We derive a central
limit theorem for the principal eigenvalue and the components of the principal
eigenvector of the adjacency matrix of a Chung-Lu random graph. Our derivation
requires certain assumptions on the average degrees that guarantee
connectivity, sparsity and bounded inhomogeneity of the graph.
