We define phylogenetic projective toric model of a trivalent graph as a
generalization of a binary symmetric model of a trivalent phylogenetic tree.
Generators of the pro- jective coordinate ring of the models of graphs with one
cycle are explicitly described. The phylogenetic models of graphs with the same
topological invariants are deforma- tion equivalent and share the same Hilbert
function. We also provide an algorithm to compute the Hilbert function.Comment: 36 pages, improved expositio