An analogue of the Gallai–Edmonds Structure Theorem for non-zero roots of the matching polynomial

Godsil observed the simple fact that the multiplicity of 0 as a root of the matching polynomial of a graph coincides with the classical notion of deficiency. From this fact he asked to what extent classical results in matching theory generalize, replacing “deficiency” with multiplicity of θ as a root of the matching polynomial. We prove an analogue of the Stability Lemma for any given root, which describes how the matching structure of a graph changes upon deletion of a single vertex. An analogue of Gallaiʹs Lemma follows. Together these two results imply an analogue of the Gallai–Edmonds Structure Theorem. Consequently, the matching polynomial of a vertex transitive graph has simple roots.