The cofactor of R[S] equals (-2)^{|S|-1} κ_G(S)/τ(G), and the normalized det R[S]/cof R[S] equals (2/|S|) tr Q + (1/2) q^T K q after Kron reduction, equivalently the max of u^T R[S] u for sum-u=1 vectors.
Devriendt, Effective resistance is more than distance: Laplacians, simplices and the Schur complement, Linear Algebra Appl
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Principal minors of effective-resistance matrices and local resistance radii
The cofactor of R[S] equals (-2)^{|S|-1} κ_G(S)/τ(G), and the normalized det R[S]/cof R[S] equals (2/|S|) tr Q + (1/2) q^T K q after Kron reduction, equivalently the max of u^T R[S] u for sum-u=1 vectors.