Strengthening the balanced set condition for the distance-regular graph of the bilinear forms

We consider a distance-regular graph $Γ=(X, \mathcal R)$ called the bilinear forms graph $H_q(D,N-D)$; we assume $N>2D\geq 6$ and $q \not=2$. We show that $Γ$ satisfies the following strengthened version of the balanced set condition. For a vertex $x \in X$ and $0 \leq i \leq D$ define $Γ_i(x)=\lbrace y \in X\vert \partial(x,y)=i\rbrace$, where $\partial$ denotes the path-length distance function. Abbreviate $Γ(x)=Γ_1(x)$. Let $V={\mathbb R}^X$ denote the standard module for ${\rm Mat}X(\mathbb R)$. For $x\in X$ let $\hat x \in V$ have $x$-coordinate 1 and all other coordinates 0. Let $E \in {\rm Mat}X(\mathbb R)$ denote the primitive idempotent that corresponds to the second largest eigenvalue of the adjacency matrix of $Γ$. For a subset $Ω\subseteq X$ define $\widehat Ω= \sum{x \in Ω} \hat x$. We fix two vertices $x,y \in X$ and write $k=\partial(x,y)$. To avoid degenerate situations, we assume $2 \leq k \leq D-1$. Using $y$ we obtain an equitable partition $\lbrace O_i \rbrace{i=1}^6$ of the local graph $Γ(x)$. By construction $O_1 = Γ(x) \cap Γ_{k-1}(y)$ and $O_6 = Γ(x) \cap Γ_{k+1}(y)$. We call $\lbrace O_i \rbrace_{i=1}^6$ the $y$-partition of $Γ(x)$. Let $\lbrace O’i \rbrace{i=1}^6$ denote the $x$-partition of $Γ(y)$. According to the original balanced set condition, for $i \in \lbrace 1,6\rbrace$ the vector $ E \widehat O_i - E \widehat O’_i$ is a scalar multiple of $E{\hat x}-E{\hat y}$. We show that for $1 \leq i \leq 6$ the vector $ E \widehat O_i - E \widehat O’_i$ is a scalar multiple of $E{\hat x}-E{\hat y}$. We investigate the consequences of this result.

💡 Research Summary

The paper studies the bilinear forms graph (H_q(D,N-D)) with parameters (N>2D\ge 6) and (q\neq2). This graph is a distance‑regular, Q‑polynomial, formally self‑dual association scheme whose vertices are the (D\times(N-D)) matrices over the finite field (\mathrm{GF}(q)) and two vertices are adjacent when their difference has rank 1. The authors focus on a refined version of the balanced set condition (BSC), a linear‑algebraic property originally introduced to characterize the Q‑polynomial property.

Main objects.
Fix two distinct vertices (x,y) with distance (k=\partial(x,y)) satisfying (2\le k\le D-1). Using the vertex (y) the authors construct an equitable partition of the local graph (\Gamma(x)) into six non‑empty cells ({O_i}_{i=1}^6); analogously they obtain the (x)-partition ({O’i}{i=1}^6) of (\Gamma(y)). The original BSC asserts that for the extreme cells (i=1,6) the vector \