We consider the infinite family of Feynman graphs known as the "banana
graphs" and compute explicitly the classes of the corresponding graph
hypersurfaces in the Grothendieck ring of varieties as well as their
Chern-Schwartz-MacPherson classes, using the classical Cremona transformation
and the dual graph, and a blowup formula for characteristic classes. We outline
the interesting similarities between these operations and we give formulae for
cones obtained by simple operations on graphs. We formulate a positivity
conjecture for characteristic classes of graph hypersurfaces and discuss
briefly the effect of passing to noncommutative spacetime.Comment: 35 pages, LaTeX, 9 eps figures (v2: small corrections