Abstract
Let G be a well covered graph, that is, all maximal independent sets of G have the same cardinality, and let ik denote the number of independent sets of cardinality k in G. We investigate the roots of the independence polynomial i(G, x) = ∑ ikxk. 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.
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Brown, J., Dilcher, K. & Nowakowski, R. Roots of Independence Polynomials of Well Covered Graphs. Journal of Algebraic Combinatorics 11, 197–210 (2000). https://doi.org/10.1023/A:1008705614290
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DOI: https://doi.org/10.1023/A:1008705614290