Weakly Hadamard diagonalizable graphs and Quantum State Transfer

Hadamard diagonalizable graphs are undirected graphs for which the corresponding Laplacian is diagonalizable by a Hadamard matrix. Such graphs have been studied in the context of quantum state transfer. Recently, the concept of a weak Hadamard matrix was introduced: a $\{-1,0, 1\}$-matrix $P$ such that $PP^T$ is tridiagonal, as well as the concept of weakly Hadamard diagonalizable graphs. We therefore naturally explore quantum state transfer in these generalized Hadamards. Given the infancy of the topic, we provide numerous properties and constructions of weak Hadamard matrices and weakly Hadamard diagonalizable graphs in order to better understand them.
Comment: 23 pages, 1 figure, 1 table