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Homometric Number of a Graph and Some Related Concepts

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Theoretical Computer Science and Discrete Mathematics (ICTCSDM 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10398))

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Abstract

Given a graph \(G=(V,E)\), two subsets \(S_1\) and \(S_2\) of the vertex set V are homometric, if their distance multisets are equal. The homometric number h(G) of a graph G is the largest integer k such that there exist two disjoint homometric subsets of cardinality k. We prove that the homometric number of the Cartesian product of two graphs is at least twice the product of the homometric numbers of the individual graphs. We also prove that the homometric number of the \(k^{th}\)-power graph of a graph G is always greater than or equal to that of G. The homometric number of some classes of graphs are also obtained. A lower bound for the homometric number of triangle-free regular graphs is obtained and two graph parameters; weak homometric number and twin number, which are related to homometric number are also discussed.

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Acknowledgments

The first author thanks University Grants Commission for granting fellowship under Faculty Development Programme (FDP).

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Authors and Affiliations

  1. Department of Mathematics, St. Peter’s College, Kolenchery, 682 311, Kerala, India

    Anu V.

  2. Department of Mathematics, St. Xavier’s College for Women, Aluva, 683 101, Kerala, India

    Aparna Lakshmanan S.

Authors
  1. Anu V.
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  2. Aparna Lakshmanan S.
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Corresponding author

Correspondence to Anu V. .

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Editors and Affiliations

  1. Kalasalingam University, Krishnankoil, India

    S. Arumugam

  2. Ball State University, Muncie, Indiana, USA

    Jay Bagga

  3. Indiana University – Purdue University, Fort Wayne, Indiana, USA

    Lowell W. Beineke

  4. Indian Institute of Technology, New Delhi, India

    B.S. Panda

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V., A., S., A.L. (2017). Homometric Number of a Graph and Some Related Concepts. In: Arumugam, S., Bagga, J., Beineke, L., Panda, B. (eds) Theoretical Computer Science and Discrete Mathematics. ICTCSDM 2016. Lecture Notes in Computer Science(), vol 10398. Springer, Cham. https://doi.org/10.1007/978-3-319-64419-6_4

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  • DOI: https://doi.org/10.1007/978-3-319-64419-6_4

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-64418-9

  • Online ISBN: 978-3-319-64419-6

  • eBook Packages: Computer ScienceComputer Science (R0)

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