Abstract
Let Γ be a distance-regular graph with diameter \(d \geqslant 3 \) and height \(h = 2 \), where \(h = max\{ i:p_{d,i}^d \ne 0\} \). Suppose that for every α in Γ and every β in \(\Gamma _d (\alpha ) \), the induced subgraph on \(\Gamma _d (\alpha ) \cap \Gamma _2 (\beta ) \) is isomorphic to a complete multipartite graph \(K_{t \times 2} \) with \(t \geqslant 2 \). Then \(d = 4 \) and Γ is isomorphic to the Johnson graph \(\begin{gathered} J(10,4) \hfill \\ \hfill \\ \end{gathered} \).
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Tomiyama, M. On Distance-Regular Graphs with Height Two, II. Journal of Algebraic Combinatorics 7, 197–220 (1998). https://doi.org/10.1023/A:1008664622576
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DOI: https://doi.org/10.1023/A:1008664622576