A classification of $Q$-polynomial distance-regular graphs with girth $6$
The electronic journal of combinatorics, Tome 32 (2025) no. 4
Let $\Gamma$ denote a $Q$-polynomial distance-regular graph with diameter $D$ and valency $k \ge 3$. In [Homotopy in $Q$-polynomial distance-regular graphs, Discrete Math., {\bf 223} (2000), 189–206], H. Lewis showed that the girth of $\Gamma$ is at most $6$. In this paper we classify graphs that attain this upper bound. We show that $\Gamma$ has girth $6$ if and only if it is either isomorphic to the Odd graph on a set of cardinality $2D +1$, or to a generalized hexagon of order $(1, k -1)$.
@article{10_37236_13897,
author = {\v{S}tefko Miklavi\v{c}},
title = {A classification of $Q$-polynomial distance-regular graphs with girth $6$},
journal = {The electronic journal of combinatorics},
year = {2025},
volume = {32},
number = {4},
doi = {10.37236/13897},
zbl = {arXiv:2501.12820},
url = {http://geodesic.mathdoc.fr/articles/10.37236/13897/}
}
Štefko Miklavič. A classification of $Q$-polynomial distance-regular graphs with girth $6$. The electronic journal of combinatorics, Tome 32 (2025) no. 4. doi: 10.37236/13897
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