Motivated by the increasing importance of large-scale
networks typically modeled by graphs we study properties associated
with the popular graph Laplacian. We exploit its mixed formulation
based on its natural factorization as product of two operators. One
result that we will report is the construction of a projection from the
edge-space onto a properly constructed coarse edge-space associated
with the edges of the coarse graph. This projection commutes with the
discrete divergence operator, and the pair of coarse edge-space and
coarse vertex-space offer the potential to construct multilevel methods
for the mixed formulation of the graph Laplacian. The performance of a
two-level method with overlapping Schwarz smoothing and correction
based on the constructed coarse spaces for solving such mixed graph
Laplacian systems is illustrated on couple of examples. We further
discuss applications using  Algebraic MultiLevel Iteration (AMLI)
Methods for graph Laplacians. The presentation is based on several
ongoing works joint with Panayot Vassilevski, Johannes Kraus, Yao
Chen, and James Brannick.