Cleavages of graphs: the spectral radius

A cleavage of a finite graph G is a morphism f: H --> G of graphs such that if P is the m x n characteristic matrix defined as P-ik = I if i is an element of f(-1) (k). otherwise = 0, then A(H)P <= PA(G). where A(G) and A(H) are the adjacency matrices of G and H, respectively. This concept generalizes induced subgraphs, quotients of graphs. Galois covers, path-tree graphs and others. We show that for spectral radii we have the inequality rho(H) <= rho(G). Equality holds only in case f: H --> G is an equivariant quotient and H has isoperimetric constant i(H) = 0.