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Christoffel–Darboux Type Identities for the Independence Polynomial

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Abstract

In this paper we introduce some Christoffel–Darboux type identities for independence polynomials. As an application, we give a new proof of a theorem of Chudnovsky and Seymour, which states that the independence polynomial of a claw-free graph has only real roots. Another application is related to a conjecture of Merrifield and Simmons.

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[1] Chihara, T. S. (2011) An Introduction to Orthogonal Polynomials, Courier Corporation.
[2] Chudnovsky, M. and Seymour, P. (2007) The roots of the independence polynomial of a clawfree graph. J. Combin. Theory Ser. B 97 350357.
[3] Godsil, C. D. (1993) Algebraic Combinatorics, Vol. 6, CRC Press.
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[5] Heilmann, O. J. and Lieb, E. H. (1972) Theory of monomer–dimer systems. Comm. Math. Phys. 25 190232.
[6] Lass, B. (2012) Mehler formulae for matching polynomials of graphs and independence polynomials of clawfree graphs. J. Combin. Theory Ser. B 102 411423.
[7] Levit, V. E. and Mandrescu, E. (2005) The independence polynomial of a graph: A survey. In 1st International Conference on Algebraic Informatics (Bozapalidis, S. et al., eds), Aristotle University of Thessaloniki, pp. 233254.
[8] Merrifield, R. and Simmons, H. (1989) Topological Methods in Chemistry, Wiley.
[9] Trinks, M. (2013) The Merrifield–Simmons conjecture holds for bipartite graphs. J. Graph Theory 72 478486.
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Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
  • URL: /core/journals/combinatorics-probability-and-computing
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