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# 数学代写|组合数学代写Combinatorial Mathematics代考|MATH069 Distance-regular graphs and P-polynomial association schemes

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## 数学代写|组合数学代写Combinatorial Mathematics代考|Distance-regular graphs and P-polynomial association schemes

In Chapter 1, Section 1.2, we defined distance-regular graphs, which are undirected connected regular graphs with parameters given in (1.10). Let $\Gamma=(X, E)$ be a distanceregular graph of diameter $d$. Let $\partial(x, y)$ be the distance between 2 vertices $x, y \in X$. For $x \in V$, define $\Gamma_i(x)={y \in X \mid \partial(x, y)=i}$. With this notation, the definition of distanceregular graphs is reformulated as follows. For any integer $i$ with $0 \leq i \leq d$ and for any $x \in X$ and $y \in \Gamma_i(x),\left|\Gamma_{i+1}(x) \cap \Gamma_1(y)\right|=b_i,\left|\Gamma_{i-1}(x) \cap \Gamma_1(y)\right|=c_i$ are constants which do not depend on the choice of $x$ and $y$, where we define $c_0=b_d=0$. In fact, if we let $k$ be the valency of a distance-regular graph, then $b_0=k$, and also if we let $z \in \Gamma_1(y)$, then the following hold:
\begin{aligned} &\partial(x, z)+\partial(z, y) \geq \partial(x, y)=i, \ &\partial(x, y)+\partial(y, z) \geq \partial(x, z) . \end{aligned}
So we have $i-1 \leq \partial(x, z) \leq i+1$, which means $z \in \Gamma_{i-1}(x) \cup \Gamma_i(x) \cup \Gamma_{i+1}(x)$. Therefore by setting $a_i=\left|\Gamma_i(x) \cap \Gamma_1(y)\right|$, we have $a_i+b_i+c_i=\left|\Gamma_1(y)\right|=k$. That is to say, $a_i$ is a constant which does not depend on the choice of $x$ and $y$. We also have $c_1=\left|\Gamma_0(x) \cap \Gamma_1(y)\right|=$ $|{x}|=1$. Moreover, since the diameter is $d$, we have $c_i>0$ for $1 \leq i \leq d$.

## 数学代写|组合数学代写Combinatorial Mathematics代考|Q-polynomial association schemes

In this section, we consider a symmetric association scheme $\mathfrak{X}$. The Bose-Mesner algebra of $\mathfrak{X}$ has a structure of a commutative algebra which is closed under the ordinary matrix product and the Hadamard product. For the algebraic structure with respect to the Hadamard product, we can define a concept similar to P-polynomial schemes.
Definition 2.80. Let $\mathfrak{X}=\left(X,\left{R_i\right}_{0 \leq i \leq d}\right)$ be a symmetric association scheme; $\mathfrak{X}$ is called a Q-polynomial association scheme or a Q-polynomial scheme with respect to the order-ing $E_0, E_1, \ldots, E_d$ if the following condition holds: For each $i(0 \leq i \leq d)$, there exists a polynomial $v_i^(x)$ of degree $i$ in the variable $x$ such that $|X| E_i=v_i^\left(|X| E_1\right)$ holds by a suitable rearrangement of the ordering of the primitive idempotents of $\mathfrak{X}$.

When we substitute a matrix into the polynomial $v_i^*(x)$, we use the Hadamard product for products of matrices.

Distance-regular graphs give a combinatorial meaning of P-polynomial schemes. Unlike P-polynomial schemes, Q-polynomial schemes do not have such combinatorial objects. However, the following proposition holds.

## 数学代写|组合数学代写组合数学代考|距离正则图和p -多项式关联方案

\begin{aligned} &\partial(x, z)+\partial(z, y) \geq \partial(x, y)=i, \ &\partial(x, y)+\partial(y, z) \geq \partial(x, z) . \end{aligned}

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