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We investigate special structures due to automorphisms in isogeny graphs ofprincipally polarized abelian varieties, and abelian surfaces in particular. Wegive theoretical and experimental results on the spectral and statisticalproperties of (2, 2)-isogeny graphs of superspecial abelian surfaces, includingstationary distributions for random walks, bounds on eigenvalues and diameters,and a proof of the connectivity of the Jacobian subgraph of the (2, 2)-isogenygraph. Our results improve our understanding of the performance and security ofsome recently-proposed cryptosystems, and are also a concrete step towards abetter understanding of general superspecial isogeny graphs in arbitrarydimension.

Automorphisms and isogeny graphs of abelian varieties, with applications to the superspecial Richelot isogeny graph