Fractional revival in complementary prisms of graphs

• URL: http://arxiv.org/abs/2609.12725v1
• Date: Fri, 11 Sep 2026 11:23:22 GMT
• Title: Fractional revival in complementary prisms of graphs
• Abstract summary: We show that for a fixed state $mathbfu$ in $G$ to the all-one vector, the complementary prism $GoverlineG$ exhibits fractional revival from the state.<n>We also characterize perfect pair state transfer in the complementary prism of a complete graph.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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