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

Abstract

We consider a distance-regular graph $\Gamma =(X,\mathcal{R})$ called the bilinear forms graph $H_q(D,N-D)$; we assume $N>2D\ge 6$ and $q\ne 2$. We show that $\Gamma$ satisfies the following strengthened version of the balanced set condition. For a vertex $x\in X$ and $0\le i\le D$ define ${\Gamma }_i(x)=\{y\in X|\partial (x,y)=i\}$, where $\partial$ denotes the path-length distance function. Abbreviate $\Gamma (x)={\Gamma }_1(x)$. Let $V={\mathbb{R}}^X$ denote the standard module for ${\mathrm{Mat}}_X(\mathbb{R})$. For $x\in X$ let $\hat{x}\in V$ have $x$-coordinate 1 and all other coordinates 0. Let $E\in {\mathrm{Mat}}_X(\mathbb{R})$ denote the primitive idempotent that corresponds to the second largest eigenvalue of the adjacency matrix of $\Gamma$. For a subset $\Omega \subseteq X$ define $\hat{\Omega }={\sum }_{x\in \Omega }\hat{x}$. We fix two vertices $x,y\in X$ and write $k=\partial (x,y)$. To avoid degenerate situations, we assume $2\le k\le D-1$. Using $y$ we obtain an equitable partition $\{O_i{\}}_{i=1}^6$ of the local graph $\Gamma (x)$. By construction $O_1=\Gamma (x)\cap {\Gamma }_{k-1}(y)$ and $O_6=\Gamma (x)\cap {\Gamma }_{k+1}(y)$. We call $\{O_i{\}}_{i=1}^6$ the $y$-partition of $\Gamma (x)$. Let $\{O_i^{\prime }{\}}_{i=1}^6$ denote the $x$-partition of $\Gamma (y)$. According to the original balanced set condition, for $i\in \{1,6\}$ the vector $E{\hat{O}}_i-E{\hat{O}}_i^{\prime }$ is a scalar multiple of $E\hat{x}-E\hat{y}$. We show that for $1\le i\le 6$ the vector $E{\hat{O}}_i-E{\hat{O}}_i^{\prime }$ is a scalar multiple of $E\hat{x}-E\hat{y}$. We investigate the consequences of this result.

36 pages