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# 数学代写|组合数学代写Combinatorial Mathematics代考|Math145 Association schemes

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## 数学代写|组合数学代写Combinatorial Mathematics代考|The deffnition of association schemes

Definition 2.1 (Association schemes). If a pair $\mathfrak{X}=\left(X,\left{R_i\right}_{0 \leq i \leq d}\right)$ of a finite set $X$ and a set $\left{R_0, R_1, \ldots, R_d\right}$ of subsets of the direct product $X \times X$ satisfies the following conditions (1), (2), (3), and (4), then $\mathfrak{X}=\left(X,\left{R_i\right}_{0 \leq i \leq d}\right)$ is called an association scheme of class $d$. In what follows, $R_i$ is called the (i-th) relation.
(1) $R_0={(x, x) \mid x \in X}$.
(2) $X \times X=R_0 \cup R_1 \cup \cdots \cup R_d$, and $R_i \cap R_j=\emptyset$ if $i \neq j$. In other words, $\left{R_0, R_1, \ldots, R_d\right}$ gives a partition of $X \times X$.
(3) Define ${ }^t R_i=\left{(x, y) \mid(y, x) \in R_i\right}$ for $R_i(0 \leq i \leq d)$. Then there exists $i^{\prime} \in{0,1, \ldots, d}$ such that ${ }^t R_i=R_{i^{\prime}}$.
(4) Fix $i, j, k \in{0,1, \ldots, d}$. Then $p_{i, j}(x, y)=\left|\left{z \in X \mid(x, z) \in R_i,(z, y) \in R_j\right}\right|$ is constant for any $(x, y) \in R_k$. In other words, the number is independent of the choice of $(x, y)$ in $R_k$, and depends only on $i, j, k$. The number is denoted by $p_{i, j}^k$ and called the intersection number.

Moreover, if the following condition holds, $\mathfrak{X}=\left(X,\left{R_i\right}_{0 \leq i \leq d}\right)$ is called a commutative association scheme.
(5) (Commutativity) For any $i, j, k \in{0,1, \ldots, d}, p_{i, j}^k=p_{j, i}^k$.
Also, if the following condition holds, $\mathfrak{X}=\left(X,\left{R_i\right}_{0 \leq i \leq d}\right)$ is called a symmetric association scheme.
(6) (Symmetry) For all $i \in{0,1, \ldots, d},{ }^t R_i=R_i$, (i. e., $i^{\prime}=i$ ).

## 数学代写|组合数学代写Combinatorial Mathematics代考|Bose–Mesner algebras

We start with the notation. For a finite set $X$, we consider $|X| \times|X|$-matrices whose rows and columns are indexed by the elements of $X$. Let $M_X(\mathbb{C})$ be the full matrix algebra of such matrices over the complex field. For $x, y \in X, M(x, y)$ denotes the $(x, y)$-entry of a matrix $M \in M_X(\mathbb{C})$. Let $I$ be the identity matrix of $M_X(\mathbb{C})$, and let $J$ be the matrix in $M_X(\mathbb{C})$ whose entries are all 1 . For a matrix $M \in M_X(\mathbb{C}),{ }^t M$ denotes the transpose of $M$. For any matrices $M_1, M_2$ in $M_X(\mathbb{C})$, we define the Hadamard product $M_1 \circ M_2$ by
$$\left(M_1 \circ M_2\right)(x, y)=M_1(x, y) M_2(x, y), \quad(x, y) \in X \times X .$$
Namely, the Hadamard product is the entry-wise product of matrices. (In elementary linear algebra, this product is forbidden.)

Let $\mathfrak{X}=\left(X,\left{R_i\right}_{0 \leq i \leq d}\right)$ be an association scheme. For each relation $R_i(0 \leq i \leq d)$, we define the matrix $A_i \in M_X(\mathbb{C})$ as follows:
$$A_i(x, y)= \begin{cases}1, & \text { if }(x, y) \in R_i, \ 0, & \text { if }(x, y) \notin R_i ;\end{cases}$$
$A_i$ is called the adjacency matrix of the relation $R_i$. Then by conditions (1), (2), (3), and (4) in Definition 2.1, we obtain the following conditions $\left(1^{\prime}\right),\left(2^{\prime}\right),\left(3^{\prime}\right)$, and $\left(4^{\prime}\right)$ :
(1 $\left.{ }^{\prime}\right) A_0=I$;
(2 $\left.2^{\prime}\right) A_0+A_1+\cdots+A_d=J$
$\left(3^{\prime}\right)$ for each $i(0 \leq i \leq d)$, there exists $i^{\prime} \in{0,1, \ldots, d}$ such that ${ }^t A_i=A_{i^{\prime}}$;
( $\left.4^{\prime}\right)$ for each $i, j(0 \leq i, j \leq d)$, there exist non-negative integers $p_{i, j}^k(0 \leq k \leq d)$ such that
$$A_i A_j=\sum_{k=0}^d p_{i, j}^k A_k .$$

## 数学代写|组合数学代写组合数学代考|关联方案的定义

(1) $R_0={(x, x) \mid x \in X}$.
(2) $X \times X=R_0 \cup R_1 \cup \cdots \cup R_d$，以及 $R_i \cap R_j=\emptyset$ 如果 $i \neq j$。换句话说， $\left{R_0, R_1, \ldots, R_d\right}$ 给出了 $X \times X$.
(3)定义 ${ }^t R_i=\left{(x, y) \mid(y, x) \in R_i\right}$ 为 $R_i(0 \leq i \leq d)$。那么就存在 $i^{\prime} \in{0,1, \ldots, d}$ 如此这般 ${ }^t R_i=R_{i^{\prime}}$
(4)修复 $i, j, k \in{0,1, \ldots, d}$。然后 $p_{i, j}(x, y)=\left|\left{z \in X \mid(x, z) \in R_i,(z, y) \in R_j\right}\right|$ 对于任何变量都是常数 $(x, y) \in R_k$。换句话说，数量与选择无关 $(x, y)$ 在 $R_k$，而只依赖于 $i, j, k$。这个数字用 $p_{i, j}^k$ 并被称为路口号。

(5)(交换性)对于任何$i, j, k \in{0,1, \ldots, d}, p_{i, j}^k=p_{j, i}^k$ .

(6)(对称性)对于所有$i \in{0,1, \ldots, d},{ }^t R_i=R_i$，(即，$i^{\prime}=i$).

.
(5)(交换性)对于所有 .
(6)(对称性)对于所有，(即，). .
(5)(交换性)对于任何 .
(6)(对称性)对于所有，(即，).

## 数学代写|组合数学代写combinatormathematics代考| Bose-Mesner algebras

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$$\left(M_1 \circ M_2\right)(x, y)=M_1(x, y) M_2(x, y), \quad(x, y) \in X \times X .$$

$$A_i(x, y)= \begin{cases}1, & \text { if }(x, y) \in R_i, \ 0, & \text { if }(x, y) \notin R_i ;\end{cases}$$
$A_i$称为关系$R_i$的邻接矩阵。然后通过定义2.1中的条件(1)(2)(3)(4)，我们得到了以下条件$\left(1^{\prime}\right),\left(2^{\prime}\right),\left(3^{\prime}\right)$和$\left(4^{\prime}\right)$:
(1 $\left.{ }^{\prime}\right) A_0=I$;
(2 $\left.2^{\prime}\right) A_0+A_1+\cdots+A_d=J$
$\left(3^{\prime}\right)$对于每个$i(0 \leq i \leq d)$，存在$i^{\prime} \in{0,1, \ldots, d}$使得${ }^t A_i=A_{i^{\prime}}$;
($\left.4^{\prime}\right)$对于每个$i, j(0 \leq i, j \leq d)$，存在非负整数$p_{i, j}^k(0 \leq k \leq d)$使得
$$A_i A_j=\sum_{k=0}^d p_{i, j}^k A_k .$$

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