Orientations, Lattice Polytopes, and Group Arrangements II: Modular and Integral Flow Polynomials of Graphs

We study modular and integral flow polynomials of graphs by means of subgroup arrangements and lattice polytopes. We introduce an Eulerian equivalence relation on orientations, flow arrangements, and flow polytopes; and we apply the theory of Ehrhart polynomials to obtain properties of modular and integral flow polynomials. The emphasis is on the geometrical treatment through subgroup arrangements and Ehrhart polynomials. Such viewpoint leads to a reciprocity law for the modular flow polynomial, which gives rise to an interpretation on the values of the modular flow polynomial at negative integers, and answers a question by Beck and Zaslavsky.