  The load-flow equations are the main tool to operate and plan electrical
networks. For transmission or distribution networks these equations can be
simplified into a linear system involving the graph Laplacian and the power
input vector. We show, using spectral graph theory, how to solve this system
efficiently. This spectral approach gives a new geometric view of the network
and power vector. This formulation yields a Parseval-like relation for the
$L_2$ norm of the power in the lines. Using this relation as a guide, we show
that a small number of eigenvector components of the power vector are enough to
obtain an estimate of the solution. This would allow fast reconfiguration of
networks and better planning.
