图的互补棱柱中的分数复兴
Fractional revival in complementary prisms of graphs
- National Institute of Technology Rourkela（印度鲁尔基拉国家理工学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出一个研究块结构实对称矩阵上分数复兴的通用框架，证明图的互补棱柱在特定条件下具有分数复兴，并刻画了完全图与完全二部图的互补棱柱中的完美对态转移。
AI中文摘要：
图 $G$ 的互补棱柱 $G\overline{G}$ 由 $G$ 与其补图 $\overline{G}$ 的不交并，并在 $G$ 中每个顶点 $a$ 与其在 $\overline{G}$ 中的副本 $a'$ 之间添加一条边而得到。本文探讨了一个用于研究具有块结构的实对称矩阵上的分数复兴的一般框架。该框架随后被用于证明：对于 $G$ 中一个与全一向量正交的固定状态 $\mathbf{u}$，互补棱柱 $G\overline{G}$ 关于邻接矩阵、拉普拉斯矩阵和无符号拉普拉斯矩阵，从状态 $[\mathbf{u},\mathbf{0}]^T$ 表现出分数复兴。我们进一步刻画了完全图的互补棱柱中的完美对态转移，并建立了完全二部图的互补棱柱中完美对态和加态转移的存在性。
英文摘要：
The complementary prism $G\overline{G}$ of a graph $G$ is obtained from the disjoint union of $G$ and its complement $\overline{G}$ by adding an edge between each vertex $a$ in $G$ and its copy $a'$ in $\overline{G}.$ This paper explores a general framework for studying fractional revival with respect to real symmetric matrices with a block structure. The framework is then used to show that, for a fixed state $\mathbf{u}$ in $G$ orthogonal to the all-one vector, the complementary prism $G\overline{G}$ exhibits fractional revival from the state $[\mathbf{u},\mathbf{0}]^T$ with respect to the adjacency, Laplacian, and signless Laplacian matrices. We further characterize perfect pair state transfer in the complementary prism of a complete graph and establish the existence of perfect pair and plus state transfer in the complementary prism of a complete bipartite graph.
