I am trying to prove that there exists only one $4$-regular connected graph with $9$ vertices with independence polynomial $1+9x+18x^2+9x^3$.Now, I am done with existence and also shown each vertex will contribute $3$, $3$-independent sets only.
I am stuck now to show that if there exists another such graph then it will be isomorphic to the obtained graph.
How should I proceed?
I ran some code in Maple and your conjecture is true. Of the 2601 connected graphs with that independence polynomial, yours is the only one that is regular. For this small example, exhaustive computer search worked well, but as far as how to proceed for other examples, I would suggest looking at this paper:
https://www.mscs.dal.ca/~hoshino/ipunique.pdf
The graph in question actually appears in Figure 1, it is $C_{9,\{3,4\}}$. I'm not sure what your bigger research question is, but circulant graphs may be of interest to you.