  We explore the dynamics of graph maps with zero topological entropy. It is
shown that a continuous map $f$ on a topological graph $G$ has zero topological
entropy if and only if it is locally mean equicontinuous, that is the dynamics
on each orbit closure is mean equicontinuous. As an application, we show that
Sarnak's M\"obius Disjointness Conjecture is true for graph maps with zero
topological entropy. We also extend several results known in interval dynamics
to graph maps. We show that a graph map has zero topological entropy if and
only if there is no $3$-scrambled tuple if and only if the proximal relation is
an equivalence relation; a graph map has no scrambled pairs if and only if it
is null if and only if it is tame.
