The Zero-Free Intervals for Characteristic Polynomials of Matroids

Authors:
Hugh Edwards;Robert Hierons;Bill Jackson
Affiliations:
Department of Mathematical and Computing Sciences, Goldsmiths College, London SE14 6NW, UK (e-mail: b.jackson@gold.ac.uk);Department of Mathematical and Computing Sciences, Goldsmiths College, London SE14 6NW, UK (e-mail: b.jackson@gold.ac.uk);Department of Mathematical and Computing Sciences, Goldsmiths College, London SE14 6NW, UK (e-mail: b.jackson@gold.ac.uk)
Venue:
Combinatorics, Probability and Computing
Year:
1998

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Abstract

Let M be a loopless matroid with rank r and c components. Let P(M, t) be the characteristic polynomial of M. We shall show that (−1)rP(M, t)≥(1−t)r for t∈(−∞, 1), that the multiplicity of the zeros of P(M, t) at t=1 is equal to c, and that (−1)r+cP(M, t)≥(t−1)r for t∈(1, 32/27]. Using a result of C. Thomassen we deduce that the maximal zero-free intervals for characteristic polynomials of loopless matroids are precisely (−∞, 1) and (1, 32/27].