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The Domination Polynomial of a Graph at −1

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Abstract

Let G be a simple graph. The domination polynomial of a graph G of order n is the polynomial \({D(G,x)=\sum_{i=0}^{n} d(G,i) x^{i}}\) , where d(G, i) is the number of dominating sets of G of size i. In this article we investigate the domination polynomial at −1. We give a construction showing that for each odd number n there is a connected graph G with D(G, −1) = n.

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References

  1. Akbari S., Alikhani S., Peng Y.H.: Characterization of graphs using domination polynomial. Europ. J. Combinatorics. 31, 1714–1724 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  2. Akbari, S., Oboudi, M.R.: Cycles are determined by their domination polynomials. Ars. Combin. (in press). Available at http://arxiv.org/abs/0908.3305

  3. Alikhani, S., Peng, Y.H.: Introduction to domination polynomial of a graph. Ars Combin (in press). Available at http://arxiv.org/abs/0905.2251

  4. Alikhani S., Peng Y.H.: Independence roots and independence fractals of certain graphs. J. Appl. Math. Comput. 36(1–2), 89–100 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. Alikhani S., Peng Y.H.: Dominating sets and domination polynomials of certain graphs. II Opuscula Mathematica. 30(1), 37–51 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Alikhani, S., Peng, Y.H.: Dominating sets and domination polynomials of paths. Int. J. Math. Math. Sci. 2009(2009), Article ID 54204

  7. Alikhani S., Peng Y.H.: Domination polynomials of cubic graphs of order 10. Turk. J. Math. 35, 355–366 (2011)

    MathSciNet  MATH  Google Scholar 

  8. Arocha J.L., Llano B.: Mean value for the matching and dominating polynomial. Discuss. Math. Graph Theory. 20, 57–69 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  9. Balister P.N., Bollobás B., Cutler J., Pebody L.: The interlace polynomial of graphs at −1. Europ. J. Combinatorics. 23, 761–767 (2002)

    Article  MATH  Google Scholar 

  10. Brouwer, A.E.: The number of dominating sets of a finite graph is odd (preprint)

  11. Frucht R., Harary F.: On the corona of two graphs. Aequationes Math. 4, 322–324 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  12. Levit, V.E., Mandrescu, E.: The independence polynomial of a graph at −1. Availabe at http://arxiv.org/abs/0904.4819v1

  13. Stanley R.P.: Acyclic orientations of graphs. Dis. Math. 5, 171–178 (1973)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Yazd University, 89195-741, Yazd, Iran

    Saeid Alikhani

  2. School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran

    Saeid Alikhani

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  1. Saeid Alikhani
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Correspondence to Saeid Alikhani.

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Alikhani, S. The Domination Polynomial of a Graph at −1. Graphs and Combinatorics 29, 1175–1181 (2013). https://doi.org/10.1007/s00373-012-1211-x

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  • DOI: https://doi.org/10.1007/s00373-012-1211-x

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