  We introduce two graph polynomials and discuss their properties. One is a
polynomial of two variables whose investigation is motivated by the performance
analysis of the Bethe approximation of the Ising partition function. The other
is a polynomial of one variable that is obtained by the specialization of the
first one. It is shown that these polynomials satisfy deletion-contraction
relations and are new examples of the V-function, which was introduced by Tutte
(1947, Proc. Cambridge Philos. Soc. 43, 26-40). For these polynomials, we
discuss the interpretations of special values and then obtain the bound on the
number of sub-coregraphs, i.e., spanning subgraphs with no vertices of degree
one. It is proved that the polynomial of one variable is equal to the
monomer-dimer partition function with weights parameterized by that variable.
The properties of the coefficients and the possible region of zeros are also
discussed for this polynomial.
