Nonexistence of a Class of Distance-Regular Graphs

Yu-pei Huang, Yeh-jong Pan, Chih-wen Weng

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Let $\Gamma$ denote a distance-regular graph with diameter $D \geq 3$ and intersection numbers $a_1=0, a_2 \neq 0$, and $c_2=1$. We show a connection between the $d$-bounded property and the nonexistence of parallelograms of any length up to $d+1$. Assume further that $\Gamma$ is with classical parameters $(D, b, \alpha, \beta)$, Pan and Weng (2009) showed that $(b, \alpha, \beta)= (-2, -2, ((-2)^{D+1}-1)/3).$ Under the assumption $D \geq 4$, we exclude this class of graphs by an application of the above connection.


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Distance-regular graph; classical parameters; parallelogram; strongly closed subgraph; D-bounded.

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