Edge-matching graph contractions and their interlacing properties

For a given graph G of order n with m edges, and a real symmetric matrix associated to the graph, M(G)∈R^n×n, the interlacing graph reduction problem is to find a graph G_r of order r<n such that the eigenvalues of M(G_r) interlace the eigenvalues of M(G). Graph contractions over partitions of the vertices are widely used as a combinatorial graph reduction tool. In this study, we derive a graph reduction interlacing theorem based on subspace mappings and the minmax theory. We then define a class of edge-matching graph contractions and show how two types of edge-matching contractions provide Laplacian and normalized Laplacian interlacing. An O(mn) algorithm is provided for finding a normalized Laplacian interlacing contraction and an O(n²+nm) algorithm is provided for finding a Laplacian interlacing contraction.