Skip to main content
Log in

On the Range of Possible Integrities of Graphs G(n, k)

  • Original Paper
  • Published:
Graphs and Combinatorics Aims and scope Submit manuscript

Abstract

We discuss the range of values for the integrity of a graphs G(n, k) where G(n, k) denotes a simple graph with n vertices and k edges. Let I max(n, k) and I min(n, k) be the maximal and minimal value for the integrity of all possible G(n, k) graphs and let the difference be D(n, k) = I max(n, k) − I min(n, k). In this paper we give some exact values and several lower bounds of D(n, k) for various values of n and k. For some special values of n and for s < n 1/4 we construct examples of graphs G n  = G n (n, n + s) with a maximal integrity of I(G n ) = I(C n ) + s where C n is the cycle with n vertices. We show that for k = n 2/6 the value of D(n, n 2/6) is at least \({\frac{\sqrt{6}-1}{3}n}\) for large n.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Atici M.: Integrity of regular graphs and integrity graphs. J. Comb. Math. Comb. Comput. 37, 27–42 (2001)

    MATH  MathSciNet  Google Scholar 

  2. Atici M., Crawford R., Ernst C.: The integrity of small cage graphs. Australas. J. Comb. 43, 39–55 (2009)

    MATH  MathSciNet  Google Scholar 

  3. Atici M., Crawford R., Ernst C.: New upper bounds for the integrity of cubic graphs. Int. J. Comput. Math. 81(11), 1341–1348 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bagga K.S., Beineke L.W., Goddard W.D., Lipman M.J., Pippert R.E.: A survey of integrity. Discrete Appl. Math. 37/38, 13–28 (1992)

    Article  MathSciNet  Google Scholar 

  5. Barefoot C.A., Entringer R., Swart H.: Vulnerability in graphs—a comparative survey. J. Comb. Math. Comb. Comput. 1, 12–22 (1987)

    MathSciNet  Google Scholar 

  6. Barefoot C.A., Entringer R., Swart H.: Integrity of trees and powers of cycles. Congr. Numer. 58, 103–114 (1987)

    MathSciNet  Google Scholar 

  7. Benko D., Ernst C., Lanphier D.: Asymptotic bounds on the integrity of graphs and separator theorems for graphs. Discrete Math. 23(1), 265–277 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  8. Goddard W., Swart H.C.: The integrity of a graph: bounds and basics. J. Comb. Math. Comb. Comput. 7, 13–151 (1990)

    MathSciNet  Google Scholar 

  9. Li, F., Li, X.: On the integrity of graphs. In: Proceedings of the 16th IASTED International Conference on Parallel and Distributed Computing and Systems, pp. 577–582 (2004)

  10. Murty U.S.R.: A generalization of the the Hoffman-Singleton graph. Ars Comb. 7, 191–193 (1979)

    MATH  Google Scholar 

  11. Vince A.: The integrity of a cubic graph. Discrete Appl. Math. 140, 223–239 (2004)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Claus Ernst.

Additional information

Claus Ernst was supported by NSF Grant #DMS-0310562.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Atici, M., Ernst, C. On the Range of Possible Integrities of Graphs G(n, k). Graphs and Combinatorics 27, 475–485 (2011). https://doi.org/10.1007/s00373-010-0990-1

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00373-010-0990-1

Keywords

Navigation