2025/12/13 by Terwilliger, Paul, Williford, Jason
#05E30 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2512.12125
We consider a type of distance-regular graph Γ=(X, \mathcal R) called a bilinear forms graph. We assume that the diameter D of Γ is at least 3. Fix adjacent vertices x,y ∈ 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 Γ and the dual Bose-Mesner algebra of Γ 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.