  Upsilon is a homomorphism on the smooth concordance group of knots defined by
Ozsv\'{a}th, Stipsicz and Szab\'{o}. In this paper, we define a generalization
of upsilon for a family of embedded graphs in rational homolog spheres. We show
that our invariant will induce a homomorphism on the homology cobordism group
of homology cylinders, and present some applications. To define this invariant,
we use tangle Floer homology. We lift relative gradings on tangle Floer
homology to absolute gradings (for certain tangles) and prove a concatenation
formula for it.
