Number of neighbors of a set of vertices in a graph

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Hi all, I'd like to know how to get the number of vertices that are adjacent to a given set of vertices in a graph. I have the following skeleton:

from sage.graphs.independent_sets import IndependentSets
G=[some graph]
J=IndependentSets(G)

And I would like to know the number of neighbors of x for each x in J (i.e., the number of vertices of G\x that are adjacent to some vertex in x). Ideally I would like something like:

F=0
t=var('t')
for x in J:
     N=number_of_neighbors(x)
     F += t^N
F

If G is a four cycle then number_of_neighbors(x)=2 for any subset x of two vertices of G, and the polynomial F above should be 1+6t^2 (because there is the empty independent set, 4 independent sets of size 1 each with 2 neighbors, and 2 independent sets of size 2 each with 2 neighbors). I appreciate your help!