For a network G, we introduce a non-singular symmetric matrix, called a path intersection matrix, that will provide a new method for computing the ratio k(G)/k(G/ab) where k(G) is the tree-number of G and G lab is obtained from GUab by contracting the new edge ab between two distinct nodes a and b. The quantities k(G)/k(G/ab) appear as invariants for various networks such as effective conductance for an electrical network and an ingredient for information centrality for a social network. We will review several examples of networks where path intersection matrices can be applied. (C) 2017 Elsevier Inc. All rights reserved.