The Forest Metrics for Graph Vertices

We propose a new graph metric and study its properties. In contrast to the standard geodesic distance in connected graphs, it takes into account all paths between vertices. Simple formulas for the increments of forest distances under basic transformations of a weighted multigraph are presented and interpreted. The notion of a cumulative weight of connections between two vertices is proposed. An interpretation for the forest distance in terms of choosing an “unsuccessful connection” between vertices is given. The reasoning involves a reciprocity principle for weighted multigraphs. Connections between the resistance distance and the forest distances are established.