Recurrence relations for the spectrum polynomial of a matroid Original Research Article

Combinatorial Laplace operators on the simplicial complex of independent sets of a loopless matroid M are known to have non-negative integral spectra (J. Amer. Math. Soc. B 13 (2000) 129). The spectrum polynomial of M, a polynomial in two variables formulated via the flats of M, is a generating function for the spectra of these operators. In this paper, we establish recurrence formulas for the spectrum polynomial of a matroid, analogous to the deletion–contraction recursions for the Tutte polynomial. However, we show that for any matroid M and e∈M the new formulas for the spectrum polynomial depend on whether or not e is closed in M. In particular, the spectrum polynomial is a new invariant for matroids that is not a Tutte–Grothendieck invariant.