### Abstract

Original language | English |
---|---|

Pages (from-to) | 2692-2710 |

Journal | Linear Algebra and its Applications |

Volume | 439 |

Issue number | 9 |

Publication status | Published - 2013 |

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### Cite this

*Linear Algebra and its Applications*,

*439*(9), 2692-2710.

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*Linear Algebra and its Applications*, vol. 439, no. 9, pp. 2692-2710.

**Geometric aspects of 2-walk-regular graphs.** / Camara Vallejo, M.; van Dam, E.R.; Koolen, J.H.; Park, J.

Research output: Contribution to journal › Article › Scientific › peer-review

TY - JOUR

T1 - Geometric aspects of 2-walk-regular graphs

AU - Camara Vallejo, M.

AU - van Dam, E.R.

AU - Koolen, J.H.

AU - Park, J.

PY - 2013

Y1 - 2013

N2 - A t-walk-regular graph is a graph for which the number of walks of given length between two vertices depends only on the distance between these two vertices, as long as this distance is at most t. Such graphs generalize distance-regular graphs and t-arc-transitive graphs. In this paper, we will focus on 1- and in particular 2-walk-regular graphs, and study analogues of certain results that are important for distance-regular graphs. We will generalize Delsarte’s clique bound to 1-walk-regular graphs, Godsil’s multiplicity bound and Terwilliger’s analysis of the local structure to 2-walk-regular graphs. We will show that 2-walk-regular graphs have a much richer combinatorial structure than 1-walk-regular graphs, for example by proving that there are finitely many non-geometric 2-walk-regular graphs with given smallest eigenvalue and given diameter (a geometric graph is the point graph of a special partial linear space); a result that is analogous to a result on distance-regular graphs. Such a result does not hold for 1-walk-regular graphs, as our construction methods will show.

AB - A t-walk-regular graph is a graph for which the number of walks of given length between two vertices depends only on the distance between these two vertices, as long as this distance is at most t. Such graphs generalize distance-regular graphs and t-arc-transitive graphs. In this paper, we will focus on 1- and in particular 2-walk-regular graphs, and study analogues of certain results that are important for distance-regular graphs. We will generalize Delsarte’s clique bound to 1-walk-regular graphs, Godsil’s multiplicity bound and Terwilliger’s analysis of the local structure to 2-walk-regular graphs. We will show that 2-walk-regular graphs have a much richer combinatorial structure than 1-walk-regular graphs, for example by proving that there are finitely many non-geometric 2-walk-regular graphs with given smallest eigenvalue and given diameter (a geometric graph is the point graph of a special partial linear space); a result that is analogous to a result on distance-regular graphs. Such a result does not hold for 1-walk-regular graphs, as our construction methods will show.

M3 - Article

VL - 439

SP - 2692

EP - 2710

JO - Linear Algebra and its Applications

JF - Linear Algebra and its Applications

SN - 0024-3795

IS - 9

ER -