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# 数学代写|组合学代写Combinatorics代考|CS-E4555 P- and Q-polynomial schemes derived from the core part

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## 数学代写|组合学代写Combinatorics代考|P- and Q-polynomial schemes derived from the core part

In this section, among known P-and Q-polynomial schemes $\mathfrak{X}=\left(X,\left{R_i\right}_{0 \leq i \leq d}\right)$, we give the list of those derived from the “core” P- and Q-polynomial schemes in Section 6.4.1. Depending on the way they are derived from the “core” P- and Q-polynomial schemes, the list is divided into the following 4 cases:
(a) derived from imprimitive schemes;
(b) derived from the second P- or Q-polynomial orderings;
(c) derived as the extended bipartite double or a fusion scheme;
(d) derived as a “cospectral scheme,” which has the same parameters as a “core” Pand Q-polynomial scheme $\mathfrak{X}$ but is not isomorphic to $\mathfrak{X}$.

## 数学代写|组合学代写Combinatorics代考|The case derived from imprimitive schemes

Let $\mathfrak{X}=\left(X,\left{R_i\right}_{0 \leq i \leq d}\right)$ be a $\mathrm{P}$ – and Q-polynomial scheme. If $\mathfrak{X}$ is an imprimitive association scheme, it is bipartite or antipodal as a P-polynomial scheme, and it is dual bipartite or dual antipodal as a Q-polynomial scheme (Section 6.1.3). If $\mathfrak{X}$ is bipartite as a P-polynomial scheme, then it is dual antipodal as a Q-polynomial scheme, and the bipartite half $\frac{1}{2} \mathfrak{X}$ also becomes $\mathrm{P}$ – and Q-polynomial scheme. Moreover, if $\mathfrak{X}$ is antipodal as a P-polynomial scheme, then it is dual bipartite as a Q-polynomial scheme, and the antipodal quotient scheme $\overline{\mathfrak{X}}$ also becomes a P- and Q-polynomial scheme.
Note that $\mathfrak{X}$ is bipartite if and only if $a_i=0(0 \leq i \leq d)$, and $\mathfrak{X}$ is antipodal if and only if $b_i=c_{d-i}\left(0 \leq i \leq d, i \neq\left[\frac{d}{2}\right]\right)$, where $a_i=p_{1, i}^i, b_i=p_{1, i+1}^i, c_i=p_{1, i-1}^i$ are the intersection numbers of $\mathfrak{X}$. Then the intersection numbers of the P-polynomial scheme $\frac{1}{2} \mathfrak{X}$ are $b_i^{\prime}=b_{2 i} b_{2 i+1} / c_2\left(0 \leq i \leq\left[\frac{d}{2}\right]-1\right), c_i^{\prime}=c_{2 i-1} c_{2 i} / c_2\left(1 \leq i \leq\left[\frac{d}{2}\right]\right)$. Moreover, the intersection numbers of the P-polynomial scheme $\overline{\mathfrak{X}}$ are $\tilde{b}i=b_i\left(0 \leq i \leq\left[\frac{d}{2}\right]-1\right), \tilde{c}_i=c_i$ $\left(1 \leq i \leq\left[\frac{d}{2}\right]-1\right)$, and $$\tilde{c}{\left[\frac{d}{2}\right]}= \begin{cases}\left(1+k_d\right) c_{\left[\frac{d}{2}\right]}, & \text { if } d \text { is even, } \ c_{\left[\frac{d}{2}\right]}, & \text { if } d \text { is odd, }\end{cases}$$
where $k_d=b_{\left[\underline{d}^d\right]} / c_{\left[\underline{d}_{]}\right.}$if $d$ is even.

## 数学代写|组合学代写Combinatorics代考|P- and Q-polynomial schemes derived from the core part

(a) 从原始方案导出;
(b) 源自第二个 $P$ 或 $Q$ 多项式排序;
(c) 派生为扩展二部双分或融合方案;

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