Spectral bounds for the maximum cut problem

The maximum cut problem this paper deals with can be formulated as follows. Given an undirected simple graph G = (V,E) where V and E stand for the node and edge sets respectively, and given weights assigned to the edges: (wij)ij∈E , a cut δ(S), with S ⊆ V is de ned as the set of edges in E with exactly one endnode in S, i.e. δ(S) = {ij ∈ E | |S ∩ {i, j}| = 1}. The weight w(S) of the cut δ(S) is the sum of the weights of the edges it contains: w(S) = ∑ ij∈δ(S) wij . The problem is then to nd a cut with maximum weight.