## Abstract

Let Γ be a distance-regular graph with valency k and diameter D, and let x be a vertex of Γ. We denote by k_{i}(0≤i≤D) the number of vertices at distance i from x. In this paper, we try to quantify the difference between antipodal and non-antipodal distance-regular graphs. We will look at the sum k_{D−1}+k_{D}, and consider the situation where k_{D−1}+k_{D}≤2k. If Γ is an antipodal distance-regular graph, then k_{D−1}+k_{D}=k_{D}(k+1). It follows that either k_{D}=1 or the graph is non-antipodal. And for a non-antipodal distance-regular graph, it was known that k_{D}(k_{D}−1)≥k and k_{D−1}≥k both hold. So, this paper concerns on obtaining more detailed information on the number of vertices for a non-antipodal distance-regular graph. We first concentrate on the case where the diameter equals three. In this case, the condition k_{D}+k_{D−1}≤2k is equivalent to the condition that the number of vertices is at most 3k+1. And we extend this result to all diameters. We note that although the result of the diameter 3 case is a corollary of the result of all diameters, the main difficulty is the diameter 3 case, and that the diameter 3 case confirms the following conjecture: there is no primitive distance-regular graph with diameter 3 having the M-property.

Original language | English |
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Pages (from-to) | 550-561 |

Number of pages | 12 |

Journal | Discrete Mathematics |

Volume | 340 |

Issue number | 3 |

DOIs | |

State | Published - 1 Mar 2017 |

## Keywords

- 2-walk-regular graphs
- Antipodal distance-regular graphs
- Distance-regular graphs
- The M-property

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