X iv : m at h . C O / 0 60 25 73 v 1 2 5 Fe b 20 06 The Forest Metric for Graph Vertices ⋆

We propose a new graph metric and study its properties. In contrast to the standard distance in connected graphs [5], it takes into account all paths between vertices. Formally, it is defined as d(i, j) = qii+qjj−qij−qji [11], where qij is the (i, j)-entry of the relative forest accessibility matrix Q(ε) = (I + εL)−1, L is the Laplacian matrix of the (weighted) (multi)graph, and ε is a positive parameter. By the matrix-forest theorem, the (i, j)-entry of the relative forest accessibility matrix of a graph provides the specific number of spanning rooted forests such that i and j belong to the same tree rooted at i. Extremely simple formulas express the modification of the proposed distance under the basic graph transformations. We give a topological interpretation of d(i, j) in terms of the probability of unsuccessful linking i and j in a model of random links. The properties of this metric are compared with those of some other graph metrics [13,1]. An application of this metric is related to clustering procedures such as centered partition [2]. In another procedure, the relative forest accessibility and the corresponding distance serve to choose the centers of the clusters and to assign a cluster to each non-central vertex. Some related geometric representations are discussed in [3]. The notion of cumulative weight of connections between two vertices is proposed. The reasoning involves a reciprocity principle for weighted multigraphs. Connections between the resistance distance and the forest distances are established.