Generating functions for modular graphs and Burgers's equation

It is shown that the generating functions of modular graphs satisfy Burgers's equations, which enable one to obtain in a unified way the generating functions for the virtual Euler characteristic and the Poincaré polynomial of the moduli space of punctured curves $\overline M_{g,n}$ and for the number (with weights $1/|{\operatorname{Aut}G}|$) of modular graphs $G$ of a definite type.