# Graph Theory and Feynman Integrals

In Vladimir A. Smirnov's book Analytic Tools for Feynman Integrals, Section 2.3, the alpha representation of general Feynman integral takes the form

$$F_{\Gamma}(q_1,\ldots,q_n;d) = \frac{i^{-a-h}\pi^{2h}}{\prod_l\Gamma(a_l)} \int_0^{\infty}\mathrm{d}\alpha_1 \ldots \int_0^{\infty}\mathrm{d}\alpha_L \prod_l\alpha_l^{a_l-1} \mathcal{U}^{-2} Z e^{i\mathcal{V}/\mathcal{U} - i\sum m_l^2\alpha_l}$$ where $$\mathcal{U}$$ and $$\mathcal{V}$$ are defined as sums running over trees and 2-trees of the given Feynman graph. I know that $$\mathcal{U}$$ is equivalent to $$\det{A}$$ in the $$4h$$-dimensional Gauss integrals, but I can't figure out how it can be expressed in the language of graph theory. Could anyone provide some help? References on the topic of graph theory and Feynman integrals are also desired.

What you need is Kirchhoff's Matrix-Tree Theorem which expresses ${\rm det}\ A$ as a sum of trees. You can find an easy "Fermionic" proof of this theorem and a list of original references in my article "The Grassmann-Berezin calculus and theorems of the matrix-tree type" (arXiv version here if you do not have access to the journal).