A contraction theorem for the largest eigenvalue of a multigraph

Let G be a multigraph with loops, and let e be an edge in G. Let H be the multigraph obtained by contracting along the edge e. Let λG and λH be the largest eigenvalues of G and H respectively. A characterisation theorem is given of precisely when λH<λG, λH=λG, or λH>λG. In the case where H happens to be a simple graph, then so is G, and the theorem subsumes those of Hoffman-Smith and Gumbrell for subdivision of edges or splitting of vertices of a graph.