  This work presents a new method for symmetrization of directed graphs that
constructs an undirected graph with equivalent pairwise effective resistances
as a given directed graph. Consequently a graph metric, square root of
effective resistance, is preserved between the directed graph and its
symmetrized version. It is shown that the preservation of this metric allows
for interpretation of algebraic and spectral properties of the symmetrized
graph in the context of the directed graph, due to the relationship between
effective resistance and the Laplacian spectrum. Additionally, Lyapunov theory
is used to demonstrate that the Laplacian matrix of a directed graph can be
decomposed into the product of a projection matrix, a skew symmetric matrix,
and the Laplacian matrix of the symmetrized graph. The application of effective
resistance preserving graph symmetrization is discussed in the context of
spectral graph partitioning and Kron reduction of directed graphs.
