  To a weighted graph can be associated a bipartite graph planar algebra P. We
construct and study the symmetric enveloping inclusion of P. We show that this
construction is equivariant with respect to the automorphism group of P. The
automorphism group of the weighted graph acts on P. We consider subgroups G of
the automorphism group of the weighted graph such that the G-fixed point space
P^G is a subfactor planar algebra. As an application we show that if G is
amenable, then P^G is amenable as a subfactor planar algebra. We define the
notions of a cocycle action of a Hecke pair on a tracial von Neumann algebra
and the corresponding cross product. We show that a large class of symmetric
enveloping inclusions of subfactor planar algebras can be described by such a
cross product.
