  We consider weighted, directed graphs with a notion of absorption on the
vertices, related to absorbing random walks on graphs. We define a generalized
inverse of the graph Laplacian, called the absorption inverse, that reflects
both the graph structure as well as the absorption rates on the vertices.
Properties of this generalized inverse are presented, including a matrix forest
theorem relating this generalized inverse to spanning forests of a related
graph, as well as relationships between the absorption inverse and the
fundamental matrix of the absorbing random walk. Applications of the absorption
inverse for describing the structure of graphs with absorption are presented,
including a directed distance metric, spectral partitioning algorithm, and
centrality measure.
