We consider a type of distance-regular graph \(\Gamma =(X, \mathcal {R})\) called a bilinear forms graph. We assume that the diameter D of \(\Gamma \) is at least 3. Fix adjacent vertices \(x,y \in X\) . In our first main result, we introduce an equitable partition of X that has \(6D-2\) subsets and the following feature: for every subset in the equitable partition, the vertices in the subset are equidistant to x and equidistant to y. This equitable partition is called the (x, y)-partition of X. By definition, the subconstituent algebra \(T=T(x)\) is generated by the Bose-Mesner algebra of \(\Gamma \) and the dual Bose-Mesner algebra of \(\Gamma \) with respect to x. As we will see, for the (x, y)-partition of X the characteristic vectors of the subsets form a basis for a T-module \(U=U(x,y)\) . In our second main result, we decompose U into an orthogonal direct sum of irreducible T-modules. This sum has five summands: the primary T-module and four irreducible T-modules that have endpoint one. We show that every irreducible T-module with endpoint one is isomorphic to exactly one of the nonprimary summands.
