Number : Title : Lattices Generated by Join of Strongly

Let $\Gamma$ be a $d$-bounded distance-regular graph with $d\geq 3$. Suppose that $P(x)$ is a set of all strongly closed subgraphs containing $x$ and that $P(x,i)$ is 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 join 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 finite atomic lattices, and give the conditions for them both being geometric lattices. We also give the eigenpolynomial of ${\mathcal{L}}'_O(x,i).$ Lattices Generated by Join of Strongly Closed Subgraphs in d-bounded Distance-regular Graphs ∗ Jun Guo Suogang Gao 1. Math. and Inf. College, Langfang Teachers’ College, Langfang, 065000, P. R. China 2. Math. and Inf. College, Hebei Normal University, Shijiazhuang, 050016, P. R. China Abstract Let Γ be a d-bounded distance-regular graph with d ≥ 3. Suppose that P (x) is a set of all strongly closed subgraphs containing x and that P (x, i) is a subset of P (x) consisting of the elements of P (x) with diameter i. Let L′(x, i) be the set generated by the join of the elements in P (x, i). By ordering L′(x, i) by inclusion or reverse inclusion, L′(x, i) is denoted by LO(x, i) or LR(x, i). We prove LO(x, i) and LR(x, i) are both finite atomic lattices, and give the conditions for them both being geometric lattices. We also give the eigenpolynomial of LO(x, i).