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Laplacian eigenvalues and the maximum cut problem. (English) Zbl 0797.90107
Summary: We introduce and study an eigenvalue upper bound \(\varphi(G)\) on the maximum cut \(\text{mc}(G)\) of a weighted graph. The function \(\varphi(G)\) has several interesting properties that resemble the behaviour of \(\text{mc}(G)\). The following results are presented: We show that \(\varphi\) is subadditive with respect to amalgam, and additive with respect to disjoint sum and 1-sum. We prove that \(\varphi(G)\) is never worse that 1.131 \(\text{mc}(G)\) for a planar, or more generally, a weakly bipartite graph with nonnegative edge weights. We give a dual characterization of \(\varphi(G)\), and show that \(\varphi(G)\) is computable in polynomial time with an arbitrary precision.

MSC:
90C35 Programming involving graphs or networks
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