Eigenpolynomials Associated with Subspaces in $d$-Bounded

Abstract

Let $\mathrm{\Gamma }$ denote a $d$-bounded distance-regular graph with diameter $d\ge 2$. A regular strongly closed subgraph of $\mathrm{\Gamma }$ is said to be a subspace of $\mathrm{\Gamma }$. Define the empty set $\mathrm{\varnothing }$ to be the subspace with diameter $-1$ in $\mathrm{\Gamma }$. For $0\le i\le d-1$, let $\mathcal{L}(\le i)$ (resp. $\mathcal{L}(\ge i)$) denote the set of all subspaces in $\mathrm{\Gamma }$ with diameters $<i$ (resp. $\ge i$) including $\mathrm{\Gamma }$ and $\mathrm{\varnothing }$. If we define the partial order on $\mathcal{L}(\le i)$ (resp. $\mathcal{L}(\ge i)$) by reverse inclusion (resp. ordinary inclusion), then $\mathcal{L}(\le i)$ (resp. $\mathcal{L}(\ge i)$) is a poset, denoted by ${\mathcal{L}}_R(\le i)$ (resp. ${\mathcal{L}}_o(\ge i)$). In the present paper, we give the eigenpolynomials of ${\mathcal{L}}_R(\le i)$ and ${\mathcal{L}}_o(\ge i)$.