Lattices Generated by Strongly Closed Subgraphs in $d$-Bounded Distance-Regular Graphs

Suogang Gao 1, Jun Guo 1, and Wen Liu 2
1 Mathematics and Information College, Hebei Normal University, Shijiazhuang 050016, P. R. China
2 Department of Mathematics, Langfang Teachers' College, Langfang 065000, P. R. China


Abstract

Let $\Gamma$ be a $d$-bounded distance-regular graph with geometric parameters $(d,\,b,\,\alpha)$ and $d\geq 3$. Suppose that $P(x)$ is a set of strongly closed subgraphs containing $x$ and that $P(x,\,i)$ a subset of $P(x)$ consisting of the elements of $P(x)$ with diameter $i$. Let ${\mathcal{L}}(x,\,i)$ be the set generated by the intersection of the elements in $P(x,\,i)$. By ordering ${\mathcal{L}}(x,\,i)$ by inclusion or reverse inclusion, ${\mathcal{L}}(x,\,i)$ is denoted by ${\mathcal{L}}_O(x,\,i)$ or ${\mathcal{L}}_R(x,\,i)$. We prove ${\mathcal{L}}_O(x,\,i)$ and ${\mathcal{L}}_R(x,\,i)$ are both nite atomic lattices, and give the conditions that they are both geometric lattices. For the $d$-bounded distance-regular graphs with diameter $d\geq 3$, we give the eigenpolynomial of $P(x)$ by ordering $P(x)$ by inclusion or reverse inclusion.