  We introduce an invariant for trivalent fatgraph spines of a once bordered
surface, which takes values in the first homology of the surface. This
invariant is the secondary object coming from two 1-cocycles on the dual
fatgraph complex, one introduced by Morita and Penner in 2008, and the other by
Penner, Turaev, and the author in 2013. We present an explicit formula for this
invariant and investigate its properties. We also show that the mod 2 reduction
of the invariant is the difference of naturally defined two spin structures on
the surface.
