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On the rational independence roots. (English) Zbl 1232.05099
Brualdi, Richard A. (ed.) et al., Combinatorics and graphs. Selected papers based on the presentations at the 20th anniversary conference of IPM on combinatorics, Tehran, Iran, May 15–21, 2009. Dedicated to Reza Khosrovshahi on the occasion of his 70th birthday. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4865-4/pbk). Contemporary Mathematics 531, 149-157 (2010).
Summary: Let \(G\) be a simple graph of order \(n\). An independent set in a graph is a set of pairwise non-adjacent vertices. The independence polynomial of \(G\) is the polynomial \[ I(G, x)= \sum^n_{k=0} s_k x^k, \] where \(s_k\) is the number of independent sets of \(G\) of size \(k\) and \(s_0= 1\). It was proved that all roots of the independence polynomial of a claw-free graph are real. In this paper, we study those graphs whose roots of the independence polynomial are rational.
For the entire collection see [Zbl 1202.05003].

05C31 Graph polynomials
05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.)