Combinatorial Properties and the Complexity of a Max-cut Approximation

We study various properties of an eigenvalue upper bound on the max-cut problem. We show that the bound behaves in a manner similar to the max-cut for the operations of switching, vertex splitting, contraction and decomposition. It can also be adjusted for branch and bound techniques. We introduce a Gram representation of a weighted graph, in order to construct weighted graphs with pre-given eigenvalue properties. As a corollary, we prove that the decision problem as to whether the upper bound coincides with the actual value of the max-cut is NP-complete. We study the mutual relation between the max-cut and the bound on the line graphs, which allow a good approximation. We show that the ratio between the upper bound and the actual size of the max-cut is close to 9/8 for the studied classes, and for several other graphs.