Tension-flow polynomials on graphs

An orientation of a graph is acyclic (totally cyclic) if and only if it is a “positive orientation” of a nowhere-zero integral tension (flow). We unify the notions of tension and flow and introduce the so-called tension-flows so that every orientation of a graph is a positive orientation of a nowhere-zero integral tension-flow. Furthermore, we introduce an (integral) tension-flow polynomial, which generalizes the (integral) tension and (integral) flow polynomials. For every graph G, the tension-flow polynomial F_G(k₁,k₂) on G and the Tutte polynomial T_G(k₁,k₂) on G satisfy F_G(k₁,k₂)⩽T_G(k₁−1,k₂−1). We also characterize the graphs for which the inequality is sharp.