Matching polynomial μ(G,x) = Σ_{k=0}^{⌊n/2⌋} (-1)^k m_k x^{n-2k}. m_k = number of matchings of size k. Heilmann-Lieb (1972): all roots are real. For trees: μ = characteristic polynomial of adjacency matrix. Encodes complete matching information.
How to Use
Select graph:
- μ(G,x): Polynomial
- m_k: k-matchings
- Roots: All real
Heilmann-Lieb Theorem
All roots of μ(G,x) are real. This is remarkable because most graph polynomials have complex roots. For trees, roots of μ equal eigenvalues of A(G). The theorem has applications in statistical mechanics.
Counting Matchings
m_0=1, m_1=|E|, m_k = k-edge matchings. The permanent of the adjacency matrix counts perfect matchings (with multiplicity). Computing permanents is #P-hard (Valiant, 1979).
Step-by-Step Instructions
- 1Select graph.
- 2Count matchings m_k.
- 3Build polynomial.
- 4Find roots.
- 5Analyze structure.