A characterization of the Grassmann graphs

Abstract

The Grassmann graph $J_q(n,D)$ is a graph on the D-dimensional subspaces of $F_q^n$ with two subspaces being adjacent if their intersection has dimension $D-1$. Characterizing these graphs by their intersection numbers is an important step towards a solution of the classification problem for $\left(P\mathrm{and}Q\right)$-polynomial association schemes, posed by Bannai and Ito in their monograph “Algebraic Combinatorics I” (1984).

Metsch (1995) [37] showed that the Grassmann graph $J_q(n,D)$ with $D\ge 3$ is characterized by its intersection numbers except for the following two principal open cases: $n=2D$ or $n=2D+1$. Van Dam and Koolen (2005) [57] constructed the twisted Grassmann graphs with the same intersection numbers as the Grassmann graphs $J_q(2D+1,D)$ (for any prime power q and $D\ge 2$), but not isomorphic to the latter ones. This shows that characterizing the graphs in the remaining cases would require a conceptually new approach.

We prove that the Grassmann graph $J_q(2D,D)$ is characterized by its intersection numbers provided that D is large enough.