We analyze Kaneko's bound to prove that, away from the -invariant , edges of multiplicity at least three can occur in the supersingular -isogeny graph only if the base field's characteristic satisfies . Further we prove a diameter bound for , while also showing that most vertex pairs have a substantially smaller distance, in the directed case; this bound is then used in conjunction with Kaneko's bound to deduce that the distance of and in is at least one fourth of the graph's diameter if . We also study other phenomena in with Kaneko's bound and provide data to demonstrate that the resulting bounds are optimal; for one of these bounds we investigate the connection between loop multiplicities in isogeny graphs and the factorization of the `diagonal' classical modular polynomial in positive characteristic.     «

We analyze Kaneko's bound to prove that, away from the -invariant , edges of multiplicity at least three can occur in the supersingular -isogeny graph only if the base field's characteristic satisfies . Further we prove a diameter bound for , while also showing that most vertex pairs have a substantially smaller distance, in the directed case; this bound is then used in conjunction with Kaneko's bound to deduce that the distance of and in is at least one fourth of the graph's diameter if ....     »