A Construction of Distance-Regular Graphs from Subspaces in \(d\)-Bounded Distance-Regular Graphs

Abstract

Let \(\Gamma\) be a \(d\)-bounded distance-regular graph with diameter \(d \geq 3\) and with geometric parameters \((d, b, \alpha)\). Pick \(x \in V(\Gamma)\), and let \(P(x)\) be the set of all subspaces containing \(x\). Suppose \(P(x, m)\) is the set of all subspaces in \(P(x)\) with diameter \(m\), where \(1 \leq m < d\). Define a graph \(\Gamma'\) whose vertex-set is \(P(x, m)\), and in which \(\Delta_1\) is adjacent to \(\Delta_2\) if and only if \(d(\Delta_1 \cap \Delta_2) = m – 1\). We prove that \(\Gamma'\) is a distance-regular graph and compute its intersection numbers.