A graph for which the inertia bound is not tight

Abstract

The inertia bound gives an upper bound on the independence number of a graph by considering the inertia of matrices corresponding to the graph. The bound is known to be tight for graphs on 10 or fewer vertices as well as for all perfect graphs. It is natural to question whether the bound is always tight. We show that the bound is not tight for the Paley graph on 17 vertices as well as its induced subgraph on 16 vertices.

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Notes

  1. 1.

    Interesting Graphs and their Colourings, unpublished lecture notes C. Godsil (2004).

  2. 2.

    See footnote 1.

  3. 3.

    Independence number of Paley graphs-IBM Research. Data published online at http://www.research.ibm.com/people/s/shearer/indpal.html.

References

  1. 1.

    Bollobás, B.: Random Graphs. Cambridge University Press, Cambridge (2001)

    Google Scholar 

  2. 2.

    Elzinga, R.J.: The Minimum Witt Index of a Graph. PhD thesis, Queen’s University. http://hdl.handle.net/1974/682 (2007)

  3. 3.

    Elzinga, R.J., Gregory, D.A.: Weighted matrix eigenvalue bounds on the independence number of a graph. Electron. J. Linear Algebra 20, 468–489 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Godsil, C., Royle, G.: Algebraic Graph Theory. Springer, New York (2001)

    Google Scholar 

  5. 5.

    Rooney, B.: Spectral Aspects of Cocliques in Graphs. PhD thesis, University of Waterloo. http://hdl.handle.net/10012/8409 (2014)

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Correspondence to John Sinkovic.

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Sinkovic, J. A graph for which the inertia bound is not tight. J Algebr Comb 47, 39–50 (2018). https://doi.org/10.1007/s10801-017-0768-0

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Keywords

  • Inertia bound
  • Cvetković bound
  • Independence number
  • Weight matrix

Mathematics Subject Classification

  • 05C50
  • 05C69
  • 15A42