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Walk-regular divisible design graphs

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Abstract

A divisible design graph (DDG for short) is a graph whose adjacency matrix is the incidence matrix of a divisible design. DDGs were introduced by Kharaghani, Meulenberg and the second author as a generalization of \((v,k,\lambda )\)-graphs. It turns out that most (but not all) of the known examples of DDGs are walk-regular. In this paper we present an easy criterion for this to happen. In several cases walk-regularity is forced by the parameters of the DDG; then known conditions for walk-regularity lead to nonexistence results for DDGs. In addition, we construct some new DDGs, and check old and new constructions for walk-regularity. In doing so, we present and use special properties in case the classes have size two. All feasible parameter sets for DDGs on at most \(27\) vertices are examined. Existence is established in all but one case, and existence of a walk-regular DDG in all cases.

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Notes

  1. This DDG was presented for the first time at the conference IPM 20—Combinatorics 2009 in Tehran as a present for the 20th anniversary of the IPM.

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Authors and Affiliations

  1. Department of Mathematics, University of Rijeka, Rijeka, Croatia

    Dean Crnković

  2. Department of Econometrics and Operations Research, Tilburg University, Tilburg, The Netherlands

    Willem H. Haemers

Authors
  1. Dean Crnković
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  2. Willem H. Haemers
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Correspondence to Willem H. Haemers.

Additional information

This is one of several papers published in Designs, Codes and Cryptography comprising the special topic on “Finite Geometries: A special issue in honor of Frank De Clerck”.

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Crnković, D., Haemers, W.H. Walk-regular divisible design graphs. Des. Codes Cryptogr. 72, 165–175 (2014). https://doi.org/10.1007/s10623-013-9861-0

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  • DOI: https://doi.org/10.1007/s10623-013-9861-0

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