We consider a conjecture of Watanabe and Yoshida in [12] concerning the Hilbert -Kunz multiplicity of an ideal in a Cohen-Macaulay ring and provide a proof of the conjecture in the case the ring is graded.

It is well-known that the "all right" metric structure on the Dehn complex of a knot is non-positively curved, or CAT(0), if and only if the knot projection is prime and alternating. We extend these ideas to spatial graphs and introduce the idea of ψ-alternating graphs as the appropriate analog of alternating knots.