  The graph bisection problem is the problem of partitioning the vertex set of
a graph into two sets of given sizes such that the sum of weights of edges
joining these two sets is optimized. We present a semidefinite programming
relaxation for the graph bisection problem with a matrix variable of order $n$
- the number of vertices of the graph - that is equivalent to the currently
strongest semidefinite programming relaxation obtained by using vector lifting.
The reduction in the size of the matrix variable enables us to impose
additional valid inequalities to the relaxation in order to further strengthen
it. The numerical results confirm that our simplified and strengthened
semidefinite relaxation provides the currently strongest bound for the graph
bisection problem in reasonable time.
