  A permutation is said to be cycle-alternating if it has no cycle double
rises, cycle double falls or fixed points; thus each index $i$ is either a
cycle valley ($\sigma^{-1}(i)>i<\sigma(i)$) or a cycle peak
($\sigma^{-1}(i)<i>\sigma(i)$). We find Stieltjes-type continued fractions for
some multivariate polynomials that enumerate cycle-alternating permutations
with respect to a large (sometimes infinite) number of simultaneous statistics
that measure cycle status, record status, crossings and nestings along with the
parity of the indices. Our continued fractions are specializations of more
general continued fractions of Sokal and Zeng. We then introduce alternating
Laguerre digraphs, which are generalization of cycle-alternating permutations,
and find exponential generating functions for some polynomials enumerating
them. We interpret the Stieltjes--Rogers and Jacobi--Rogers matrices associated
to some of our continued fractions in terms of alternating Laguerre digraphs.
