  Let $A$ and $B$ be local operators in Hamiltonian quantum systems with $N $
degrees of freedom and finite-dimensional Hilbert space. We prove that the
commutator norm $\lVert [A(t),B]\rVert$ is upper bounded by a topological
combinatorial problem: counting irreducible weighted paths between two points
on the Hamiltonian's factor graph. Our bounds sharpen existing Lieb-Robinson
bounds by removing extraneous growth. In quantum systems drawn from zero-mean
random ensembles with few-body interactions, we prove stronger bounds on the
ensemble-averaged out-of-time-ordered correlator $\mathbb{E}\left[ \lVert
[A(t),B]\rVert_F^2\right]$. In such quantum systems on Erd\"os-R\'enyi factor
graphs, we prove that the scrambling time $t_{\mathrm{s}}$, at which $\lvert
[A(t),B]\rVert_F=\mathrm{\Theta}(1)$, is almost surely
$t_{\mathrm{s}}=\mathrm{\Omega}(\sqrt{\log N})$; we further prove
$t_{\mathrm{s}}=\mathrm{\Omega}(\log N) $ to high order in perturbation theory
in $1/N$. We constrain infinite temperature quantum chaos in the $q$-local
Sachdev-Ye-Kitaev model at any order in $1/N$; at leading order, our upper
bound on the Lyapunov exponent is within a factor of 2 of the known result at
any $q>2$. We also speculate on the implications of our theorems for
conjectured holographic descriptions of quantum gravity.
