  Symmetric edge polytopes are a class of lattice polytopes constructed from
finite simple graphs. In the present paper we highlight their connections to
the Kuramoto synchronization model in physics -- where they are called
adjacency polytopes -- and to Kantorovich--Rubinstein polytopes from finite
metric space theory. Each of these connections motivates the study of symmetric
edge polytopes of particular classes of graphs. We focus on such classes and
apply algebraic-combinatorial methods to investigate invariants of the
associated symmetric edge polytopes.
