  We investigate special structures due to automorphisms in isogeny graphs of
principally polarized abelian varieties, and abelian surfaces in particular. We
give theoretical and experimental results on the spectral and statistical
properties of (2, 2)-isogeny graphs of superspecial abelian surfaces, including
stationary distributions for random walks, bounds on eigenvalues and diameters,
and a proof of the connectivity of the Jacobian subgraph of the (2, 2)-isogeny
graph. Our results improve our understanding of the performance and security of
some recently-proposed cryptosystems, and are also a concrete step towards a
better understanding of general superspecial isogeny graphs in arbitrary
dimension.
