An Analogue of the Gallai-Edmonds Structure Theorem for Nonzero Roots of the Matching Polynomial

Details An Analogue of the Gallai-Edmonds Structure Theorem for Nonzero Roots of the Matching Polynomial An Analogue of the Gallai-Edmonds Structure Theorem for Nonzero Roots of the Matching Polynomial

2008-06-27

arxiv.org/abs/0806.4619v1

Mathematics

Combinatorics

Scientific paper

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 $\theta$ 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 Gallai's 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.

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