  We examine a class of N=2 supersymmetric gauge theories in (3+1) dimensions
whose Lagrangians are determined by graphs consisting of two building blocks,
namely a tri-vertex and a line. A line represents an SU(2) gauge group and a
tri-vertex represents a matter field in the trifundamental representation of
SU(2)^3. These graphs can be topologically classified by the genus and the
number of external legs. This paper focuses on the hypermultiplet moduli spaces
of the aforementioned theories. We compute the Hilbert series which count all
chiral operators on the hypermultiplet moduli space. Several examples show that
theories corresponding to different graphs with the same genus and the same
number of external legs possess the same Hilbert series. This is in agreement
with the conjecture that such theories are related to each other by S-duality.
We also give a general expression for the Hilbert series for the graph with any
genus and any number of external legs.
