  We show the existence of the scaling exponent $\chi\in (0,4[(1+\gamma^2/4)-
\sqrt{1+\gamma^4/16}]/\gamma^2]$ of the graph distance associated with
subcritical two-dimensional Liouville quantum gravity of paramater $\gamma<2$
on $\mathbb V =[0,1]^2 $. We also show that the Liouville heat kernel
satisfies, for any fixed $u,v\in \mathbb V^o$, the short time estimates $$
\lim_{ t \to 0} \frac{\log |\log {\mathsf p}_t^\gamma(u,v)| }{|\log
t|}=\frac{\chi}{2-\chi}, \ \mbox{\rm a.s.} $$
