Laplacian eigenvalues and the maximum cut problem

We introduce and study an eigenvalue upper boundϕ(G) on the maximum cut mc (G) of a weighted graph. The functionϕ(G) has several interesting properties that resemble the behaviour of mc (G). The following results are presented.We show thatϕ is subadditive with respect to amalgam, and additive with respect to disjoint sum and 1-sum. We prove thatϕ(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 ofϕ(G), and show thatϕ(G) is computable in polynomial time with an arbitrary precision.

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