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Diameter of Neighborhood Graphs

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Abstract

The neighborhood graph \(G'\) of a graph G has the same vertex set as G and two vertices are adjacent in \(G'\) if and only if they have a common neighbor in G. We study the diameter \(\mathrm{diam}(G')\) of the neighborhood graph \(G'\) in terms of the diameter of G. We show that if G is a connected non-bipartite graph of diameter d, then \(\lceil d/2 \rceil \le \mathrm{diam}(G') \le d\) and the bounds are best possible for every \(d \ge 1\). If G is a connected bipartite graph, then \(G'\) has 2 components. We also present results on the diameter of components of \(G'\), if \(G'\) is the neighborhood graph of a connected bipartite graph.

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References

  1. Ayyaswamy, S.K., Balachandran, S., Kannan, K.: Bounds on the second stage spectral radius of graphs. Int. J. Comput. Math. Sci. 3, 424–427 (2009)

    MathSciNet  Google Scholar 

  2. Alwardi, A., Arsić, B., Gutman, I., Soner, N.D.: The common neighborhood graph and its energy. Iran. J. Math. Sci. Inf. 7, 1–8 (2012)

    MathSciNet  MATH  Google Scholar 

  3. Alwardi, A., Soner, N.D., Gutman, I.: On the common-neighborhood energy of a graph. Bulletin, Classe des Sciences Mathématiques et Naturelles, Sciences Mathématiques 36, 49–59 (2011)

    MathSciNet  MATH  Google Scholar 

  4. Boland, J.W., Brigham, R.C., Dutton, R.D.: Embedding arbitrary graphs in neighborhood graphs. J. Comb. Syst. Sci. 12, 101–112 (1987)

  5. Boland, J.W., Brigham, R.C., Dutton, R.D.: The difference between a neighborhood graph and a wheel. Congressus Numer. 58, 151–156 (1987)

    MathSciNet  MATH  Google Scholar 

  6. Cozzens, M.: Food webs, competition graphs, and habitat formation. Math. Model. Nat. Phenom. 6, 22–38 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. Brigham, R.C., Dutton, R.D.: On neighbourhood graphs. J. Comb. Syst. Sci. 12, 75–84 (1987)

    MathSciNet  MATH  Google Scholar 

  8. Jog, S.R., Hande, S.P., Gutman, I., Bozkurt, S.B.: Derived graphs of some graphs. Kragujev. J. Math. 36, 309–314 (2012)

    MathSciNet  MATH  Google Scholar 

  9. Schiermeyer, I., Sonntag, M., Teichert, H.-M.: Structural properties and Hamiltonicity of neighborhood graphs. Gr. Comb. 26, 433–456 (2010)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The work was supported by the National Research Foundation of South Africa; Grant numbers: 91499, 90793.

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Authors and Affiliations

  1. Department of Mathematics and Applied Mathematics, University of the Free State, Bloemfontein, South Africa

    Tomáš Vetrík

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  1. Tomáš Vetrík
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Correspondence to Tomáš Vetrík.

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Communicated by Xueliang Li.

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Vetrík, T. Diameter of Neighborhood Graphs. Bull. Malays. Math. Sci. Soc. 39 (Suppl 1), 117–122 (2016). https://doi.org/10.1007/s40840-015-0231-0

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  • DOI: https://doi.org/10.1007/s40840-015-0231-0

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