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

TL;DR

The paper classifies $Q$-polynomial distance-regular graphs with diameter $D$ and girth $6$, showing the only possibilities are the Odd graph on a set of cardinality $2D+1$ or a generalized hexagon of order $(1,k-1)$. The authors split into almost bipartite and bipartite cases, using intersection numbers, eigenvalue structure, and known results (e.g., Le's girth bound, LT, Ca1) to narrow to these two families. They prove that for almost bipartite graphs, girth $6$ forces the Odd graph; for bipartite graphs with $D \ne 5$, girth $6$ forces $D=3$ and a generalized hexagon, while in the $D=5$ case, the girth is forced to be $4$. The results complete the classification and connect $Q$-polynomial graphs to classical incidence geometries, enriching understanding of the interplay between algebraic and combinatorial structure in distance-regular graphs.

Abstract

Let $Γ$ 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 $Γ$ is at most $6$. In this paper we classify graphs that attain this upper bound. We show that $Γ$ 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 $(1, k -1)$.

Theorems & Definitions (20)

• Lemma 2.1
• Theorem 3.1
• Theorem 3.2
• Corollary 3.3
• Theorem 3.4
• Lemma 3.5
• Proposition 3.6
• Theorem 4.1
• Proposition 5.1
• Proposition 5.2