3-sat and Tutte polynomial

Question

Please consider the following 3-SAT instance and the corresponding graph

The graph can be displayed in other two forms

The Tutte polynomial for this graph is

The independence number of the graph is 4, then the considered 3-SAT instance is satisfiable. This fact is checked using the code:

x1, x2, x3 = Bools('x1 x2 x3')
s=Solver()
s.add(Or(x1,x2,Not(x3)),Or(x1,Not(x2),x3),Or(Not(x1),x2,x3),Or(Not(x1),Not(x2),Not(x3)))
print s
print s.check()
m = s.model()
print m  

and the corresponding output is:

sat
[x3 = False, x2 = False, x1 = False]

The corresponding complement of the graph is

and the Tutte polynomial for the complement of the graph is

The clique number of the complement is 4 and then it says that the considered 3-SAT instance is satisfiable.

The question is : It is possible to use the Tutte polynomial to decide if the considered 3-SAT instance is satisfiable ?

Answer 1

Other example:

The Tutte polynomial is:

The independence number of the graph is 3, then the considered 3-SAT instance is satisfiable. This fact is checked using the code:

x1, x2, x3, x4 = Bools('x1 x2 x3 x4')
s = Solver()
s.add(Or(Not(x1),x2,x3),Or(x1,Not(x2),x3),Or(Not(x1),x2,x4))
print s.check()
m = s.model()
print m

and the corresponding output is:

sat
[x3 = False, x2 = False, x1 = False, x4 = False]

The corresponding complement of the graph is

and the Tutte polynomial for the complement of the graph is

The clique number of the complement is 3 and then it says that the considered 3-SAT instance is satisfiable.

Answer 2

Other example:

the graph can be displayed in other two forms:

The Tutte polynomial for this graph is:

The independence number of the graph is 3, then the considered 3-SAT instance is satisfiable. This fact is checked using the code:

x, y, z = Bools('x y z')
s = Solver()
s.add(Or(x,y,z),Or(Not(x),Not(y),Not(z)),Or(x,Not(y),Not(z)))
print s.check()
m = s.model()
print m

and the corresponding output is:

sat
[z = False, y = True, x = False]

The corresponding complement of the graph is

and the Tutte polynomial for the complement of the graph is

The clique number of the complement is 3 and then it says that the considered 3-SAT instance is satisfiable.

The question is : It is possible to use the Tutte polynomial to count the possible models for the considered satisfiable 3-SAT instance?