Frustration and isoperimetric inequalities for signed graphs

Let Gσ=(V,E,σ) be a connected signed graph. Using the equivalence between signed graphs and 2-lifts of graphs, we show that the frustration index of Gσ is bounded from below and above by expressions involving another graph invariant, the smallest eigenvalue of the (signed) Laplacian of Gσ. From the proof, stricter bounds are derived. Additionally, we show that the frustration index is the solution to a l1-norm optimization problem over the 2-lift of the signed graph. This leads to a practical implementation to compute the frustration index. Also, leveraging the 2-lifts representation of signed graphs, a straightforward proof of Harary's theorem on balanced graphs is derived. Finally, real world examples are considered.