A classification of $Q$-polynomial distance-regular graphs with girth $6$

TL;DR

This paper classifies $Q$-polynomial distance-regular graphs with girth exactly 6, showing they are either odd graphs or generalized hexagons, thus completing the characterization of such graphs with maximal girth.

Contribution

The paper provides a complete classification of $Q$-polynomial distance-regular graphs with girth 6, identifying them as either odd graphs or generalized hexagons.

Findings

Graphs with girth 6 are either odd graphs or generalized hexagons.

Girth of $Q$-polynomial distance-regular graphs is at most 6.

Classification completes understanding of graphs with maximal girth.

Abstract

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 $(1, k -1)$.