## Abstract

In this paper, we look at distance-regular graphs with induced subgraphs K_{r}_{,}_{t}, where 1 ≤ r≤ t are integers. In particular, we show that if a distance-regular graph Γ with diameter D≥ 5 contains an induced subgraph K_{2}_{,}_{t}(t≥ 2) , then t is bounded above by a function of b1θ1+1, where θ_{1} is the second largest eigenvalue of Γ. Using this bound we obtain that the intersection number c_{2} of a μ-graph-regular distance-regular graph with diameter D≥ 5 and large a_{1} is bounded above by a fuction of b=⌈b1θ1+1⌉. We then apply this bound to thin Q-polynomial distance-regular graphs with diameter D≥ 5 and large a_{1} to show that c_{2} is bounded above by a function of ⌈b1θ1+1⌉. At last, we again apply the bound to thin distance-regular graphs with classical parameters (D, b, α, β) to show that the parameter α is bounded above by a function of b1θ1+1.

Original language | English |
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Article number | 175 |

Journal | Graphs and Combinatorics |

Volume | 38 |

Issue number | 6 |

DOIs | |

State | Published - Dec 2022 |

## Keywords

- Classical parameters
- Distance-regular graphs
- Q-Polynomial distance-regular graphs
- Thin distance-regular graphs

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