A common annihilator for resistance numerator and denominator families

Let a family of graphs have Laplacian matrices, and apply the procedures LaplaceExpand and SystemReduce together with the mathematical theorems described in the source. The numerator and denominator in Bapat's effective-resistance formula are the associated determinant sequences. Common-annihilator conjecture. For any family of graphs with specified criteria, LaplaceExpand, SystemReduce, and the mathematical theorems suffice to find a single annihilator for both the numerator and denominator of the effective-resistance formula.

For banded underlying Laplacians, the same annihilator can be obtained from Laplace expansions of the numerator sequence and also annihilates the denominator; for non-banded Laplacians, the source notes that SystemReduce may need to be applied separately. The general criteria ensuring the procedure works remain open.