With regards to your Q2, in quantum field theory there is a commonly used generalization of the Kirchhoff polynomial to 2-trees (2-component spanning forests). It's normally called the 2nd Symanzik polynomial, the first Symanzik polynomial being basically identical to the Kirchhoff polynomial. I'm not sure if it can generalize to k-spanning forests.

To calculated the 2nd Symanzik polynomial you need to associate a variable with each vertex. (In QFT this is the incoming momentum at that vertex.)

There's a nice recent review article which discusses some of this
Feynman graph polynomials (arXiv:1002.3458v3)

I also made a Mathematica demonstration that lets you draw graphs and calculates the polynomials.
Scalar Feynman Diagrams And Symanzik Polynomials.

The classic reference is
N. Nakanishi, Graph Theory and Feynman Integrals, Newark, NJ: Gordon and Breach, 1971.

With regards to your Q2, in quantum field theory there is a commonly used generalization of the Kirchhoff polynomial to 2-trees (2-component spanning forests). It's normally called the 2nd Symanzik polynomial, as the 1st Symanzik polynomial is basically identical to the Kirchhoff polynomial. I'm not sure if it can generalize to $k$-spanning forests.

To calculate the 2nd Symanzik polynomial you need to associate a variable with each vertex. (In QFT this is the incoming momentum at that vertex.)

There's a nice recent review article which discusses some of this
Feynman graph polynomials (arXiv:1002.3458v3)

I also made a Mathematica demonstration that lets you draw graphs and calculates the polynomials.
Scalar Feynman Diagrams And Symanzik Polynomials.

The classic reference is
N. Nakanishi, Graph Theory and Feynman Integrals, Newark, NJ: Gordon and Breach, 1971.

With regards to your Q2, in quantum field theory there is a commonly used generalization of the Kirchhoff (first Symanzik) polynomial to 2-trees (2-component spanning forests). It's normally called the 2nd Symanzik polynomial. I'm not sure if it can generalize to k-spanning forests.

To calculated the 2nd Symanzik polynomial you need to associate a variable with each vertex. (In QFT this is the incoming momentum at that vertex.)

There's a nice recent article which discusses some of this
Feynman graph polynomials (arXiv:1002.3458v3)

I also made a Mathematica demonstration that lets you draw graphs and calculates the polynomials.
Scalar Feynman Diagrams And Symanzik Polynomials.

The classic reference is
N. Nakanishi, Graph Theory and Feynman Integrals, Newark, NJ: Gordon and Breach, 1971.

With regards to your Q2, in quantum field theory there is a commonly used generalization of the Kirchhoff polynomial to 2-trees (2-component spanning forests). It's normally called the 2nd Symanzik polynomial, the first Symanzik polynomial being basically identical to the Kirchhoff polynomial. I'm not sure if it can generalize to k-spanning forests.

To calculated the 2nd Symanzik polynomial you need to associate a variable with each vertex. (In QFT this is the incoming momentum at that vertex.)

There's a nice recent review article which discusses some of this
Feynman graph polynomials (arXiv:1002.3458v3)

I also made a Mathematica demonstration that lets you draw graphs and calculates the polynomials.
Scalar Feynman Diagrams And Symanzik Polynomials.

The classic reference is
N. Nakanishi, Graph Theory and Feynman Integrals, Newark, NJ: Gordon and Breach, 1971.