Graphical functions in parametric space

Graphical functions are positive functions on the punctured complex plane $\mathbb{C}\setminus \{0,1\}$ which arise in quantum field theory. We generalize a parametric integral representation for graphical functions due to Lam, Lebrun and Nakanishi, which implies the real analyticity of graphical functions. Moreover we prove a formula that relates graphical functions of planar dual graphs.