The graph partition problem is the problem of partitioning the vertex set of
a graph into a fixed number of sets of given sizes such that the sum of weights
of edges joining different sets is optimized. In this paper we simplify a known
matrix-lifting semidefinite programming relaxation of the graph partition
problem for several classes of graphs and also show how to aggregate additional
triangle and independent set constraints for graphs with symmetry. We present
an eigenvalue bound for the graph partition problem of a strongly regular
graph, extending a similar result for the equipartition problem. We also derive
a linear programming bound of the graph partition problem for certain Johnson
and Kneser graphs. Using what we call the Laplacian algebra of a graph, we
derive an eigenvalue bound for the graph partition problem that is the first
known closed form bound that is applicable to any graph, thereby extending a
well-known result in spectral graph theory. Finally, we strengthen a known
semidefinite programming relaxation of a specific quadratic assignment problem
and the above-mentioned matrix-lifting semidefinite programming relaxation by
adding two constraints that correspond to assigning two vertices of the graph
to different parts of the partition. This strengthening performs well on highly
symmetric graphs when other relaxations provide weak or trivial bounds