Polynomials Associated with Nowhere-Zero Flows

In this paper we study relations between nowhere-zero Zk− and integer-valued flows in graphs and the functions FG(k) and IG(k) evaluating the numbers of nowhere-zero Zk− and k-flows in a graph G, respectively. It is known that FG(k) is a polynomial for k>0. We show that IG(k) is also a polynomial and that 2m(G)FG(k)⩾IG(k)⩾(m(G)+1) FG(k), where m(G) is the rank of the cocycle matroid of G. Finally we prove that FG(k+1)⩾FG(k)·k/(k−1) and IG(k+1)⩾ IG (k)·k/(k−1) for every k>1.