Laplacian eigenvalues and the maximum cut problem

We introduce and study an eigenvalue upper bound p(G) on the maximum cut mc (G) of a weighted graph. The function ~o(G) has several interesting properties that resemble the behaviour of mc (G). The following results are presented. We show that q~ is subadditive with respect to amalgam, and additive with respect to disjoint sum and 1-sum. We prove that ~(G) is never worse that 1.131 mc(G) for a planar, or more generally, a weakly bipartite graph with nonnegative edge weights. We give a dual characterization of ~o(G), and show that q~(G) is computable in polynomial time with an arbitrary precision.