Conformal decomposition of integral tensions and potentials of signed graphs

Let be a subgroup of an integral chain group on a set E. A nonzero chain g of E is said to be conformally decomposable if there exist nonzero chains g1; g2 of such that g = g1+g2 and g1(e)g2(e) ≥ 0 for all e 2 E. For a signed graph Σ with edge set E, there are two subgroups F(Σ; Z) and T(Σ; Z) of the 1-chain group C1(Σ; Z), known as the flow lattice and tension lattice of Σ. The conformally indecomposable flows of F(Σ; Z) are classified in [B. Chen and J.Wang, arXiv:1112.0642, 2011; B. Chen, J. Wang, and T. Zaslavsky, Discrete Math., 340 (2017), pp. 1271{1786] as signed- graphic circuit flows and a class of characteristic vectors of certain directed Eulerian cycle-trees. In this paper we classify conformally indecomposable tensions of T(Σ; Z) as characteristic vectors of signed-graphic directed bonds and a class of characteristic vectors of directed semi-bonds and directed hyper-bonds. The half-spin structures (Σ1 2 -potential functions) of Σ correspond to characteristic vectors of directed hyper-bonds. A byproduct is the classification of conformally indecomposable integral potentials.