论文标题
在锤子图中的本征函数和最低1-完整的位拉。
Eigenfunctions and minimum 1-perfect bitrades in the Hamming graph
论文作者
论文摘要
Hamming Graph $ h（n，q）$是该图，其顶点是字母$ \ \ \ {0,1，\ ldots，q-1 \} $的长度$ n $的单词，如果两个坐标在一个坐标上，则两个顶点相邻。 $ h（n，q）$的邻接矩阵具有$ n+1 $ dintival eigenvalues $ n（q-1）-q \ cdot i $，带有相应的eigenspaces $ u_ {i}（i}（n，q）$，for $ 0 \ leq i \ leq i \ leq n $。在这项工作中，我们研究了属于直接sum $ u_i（n，q）\ oplus u_ {i+1}（n，q）\ oplus \ ldots \ oplus u_j（n，q）$的函数。我们发现以$ q = 2 $和$ q = 3 $，$ i+j> n $的$ q = 2 $的支持的最低基数支持。特别是，我们发现来自特征的最小基数对特征的支持$ u_ {i}（n，3）$ for $ i> \ frac {n} {2} $。利用$ 1 $完美的位拉德和特征函数之间的对应关系与特征值$ -1 $，我们在Hamming Graph $ h（n，3）$中找到了$ 1 $ - 完美的bitrade的最小尺寸。
The Hamming graph $H(n,q)$ is the graph whose vertices are the words of length $n$ over the alphabet $\{0,1,\ldots,q-1\}$, where two vertices are adjacent if they differ in exactly one coordinate. The adjacency matrix of $H(n,q)$ has $n+1$ distinct eigenvalues $n(q-1)-q\cdot i$ with corresponding eigenspaces $U_{i}(n,q)$ for $0\leq i\leq n$. In this work we study functions belonging to a direct sum $U_i(n,q)\oplus U_{i+1}(n,q)\oplus\ldots\oplus U_j(n,q)$ for $0\leq i\leq j\leq n$. We find the minimum cardinality of the support of such functions for $q=2$ and for $q=3$, $i+j>n$. In particular, we find the minimum cardinality of the support of eigenfunctions from the eigenspace $U_{i}(n,3)$ for $i>\frac{n}{2}$. Using the correspondence between $1$-perfect bitrades and eigenfunctions with eigenvalue $-1$, we find the minimum size of a $1$-perfect bitrade in the Hamming graph $H(n,3)$.
