Flow polytopes and the graph of reflexive polytopes

We suggest defining the structure of an unoriented graph Rd on the set of reflexive polytopes of a fixed dimension d. The edges are induced by easy mutations of the polytopes to create the possibility of walks along connected components inside this graph. For this, we consider two types of mutations: Those provided by performing duality via nef-partitions, and those arising from varying the lattice. Then for d ≤ 3, we identify the flow polytopes among the reflexive polytopes of each single component of the graph Rd. For this, we present for any dimension d ≥ 2 an explicit finite list of quivers giving all d-dimensional reflexive flow polytopes up to lattice isomorphisms. We deduce as an application that any such polytope has at most 6(d− 1) facets.