Families of graphs satisfying the spectral edges conjecture

A Bloch variety of a Z^d periodic graph with m orbits of vertices
and  isolated  critical  points  has  at  least  2^d  m  critical
points.  (These  occur  over  the  corner  points,  those  points
satisfying z^2=1.)

I will describe  two families of Z^d periodic  graphs whose Bloch
varieties have  only these critical points.   Such graphs satisfy
Kuchment's  spectral edges  nondegeneracy conjecture.   The first
family  are  graphs  whose  Newton  polytope  has  base  a  cross
polytope, while the second are given in terms of the graph.

This is joint work with Matthew Faust.