We introduce and study an eigenvalue upper bound phi(G) on the maximum
cut mc (G) of a weighted graph. The function phi(G) has several inter
esting properties that resemble the behaviour of mc(G). The following
results are presented. We show that phi is subadditive with respect to
amalgam, and additive with respect to disjoint sum and 1-sum. We prov
e that phi(G) is never worse that 1. 131 mc(G) for a planar, or more g
enerally, a weakly bipartite graph with nonnegative edge weights. We g
ive a dual characterization of phi(G), and show that phi(G) is computa
ble in polynomial time with an arbitrary precision.