Fractional revival in complementary prisms of graphs

The complementary prism $G\bar{G}$ of a graph $G$ is obtained from the disjoint union of $G$ and its complement $\bar{G}$ by adding an edge between each vertex $a$ in $G$ and its copy $a^{\mathrm{\prime }}$ in $\bar{G}\mathrm{.}$ 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 $G\bar{G}$ 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.