AI 中文总结
本文刻画均匀加权图设计的存在性，给出存在与不存在此类设计的图族，证明线性码与线性正交阵列的对偶性，还构造出拉普拉斯特征多项式几乎不可约且无均匀加权设计的图族。
AI 中文摘要
图设计是图的顶点子集，每个选中顶点带有一个权重，可对图上函数的选定子空间进行完美平均。若所有权重相等，则该设计为均匀加权设计，正交阵列、组合区组设计、t次置换等多个知名组合对象均属于均匀加权图设计。尽管人们可能认为均匀加权设计会出现在结构化图中，但它们并非总是存在。本文刻画了均匀加权图设计的存在性，利用所得结果给出了若干存在和不存在此类设计的图族。我们的结果为控制这些设计的存在性及基数的结构提供了多面体视角，尤其刻画了阈值图的所有均匀加权设计，并给出了线性码与线性正交阵列对偶性的几何证明。我们还为拉普拉斯特征多项式几乎不可约的图提供了一种新构造，以生成不存在均匀加权设计的图族。
英文摘要
A graphical design is a subset of vertices of a graph, along with a weight for each chosen vertex, that can perfectly average chosen subspaces of functions on the graph. A design is uniformly weighted if all the weights are equal, and several well-known combinatorial objects such as orthogonal arrays, combinatorial block designs and t-wise permutations are uniformly weighted graphical designs. While one might expect to see uniformly weighted designs in structured graphs, they do not always exist. In this paper we characterize the existence of uniformly weighted graphical designs, and use our result to provide several families of graphs that have, and do not have, such designs. Our results offer a polyhedral view of the structures that control the existence and cardinalities of these designs. In particular, we characterize all uniformly weighted designs of threshold graphs, and provide a geometric proof of the duality of linear codes and linear orthogonal arrays. We also provide a novel construction for graphs whose Laplacian characteristic polynomials are almost irreducible, to produce families without uniformly weighted designs.
