Roots of Independence Polynomials of Well Covered Graphs

  • Authors:

  • J. I. Brown;K. Dilcher;R. J. Nowakowski

  • Affiliations:

  • Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia, Canada B3H 3J5. brown@mscs.dal.ca;Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia, Canada B3H 3J5. dilcher@mscs.dal.ca;Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia, Canada B3H 3J5. rjn@mscs.dal.ca

  • Venue:
  • Journal of Algebraic Combinatorics: An International Journal
  • Year:
  • 2000

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Abstract

Let G be a well covered graph, that is, all maximal independent sets of G have the same cardinality, and let i_k denote the number of independent sets of cardinality k in G. We investigate the roots of the independence polynomial i(G, x) = ∑ i_kx^k. In particular, we show that if G is a well covered graph with independence number β, then all the roots of i(G, x) lie in in the disk |z| ≤ β (this is far from true if the condition of being well covered is omitted). Moreover, there is a family of well covered graphs (for each β) for which the independence polynomials have a root arbitrarily close to −β.