Weakly Hadamard diagonalizable graphs

Abstract

A matrix is called weakly Hadamard if its entries are from {0, -1, 1} and its non-consecutive columns (with some ordering) are orthogonal. Unlike Hadamard matrices, there is a weakly Hadamard matrix of order n for every n>= 1. In this work, graphs for which their Laplacian matrices can be diagonalized by a weakly Hadamard matrix are studied. A number of necessary and sufficient conditions are verified along with identification of numerous families of graphs whose Laplacian matrices can be diagonalized by a weakly Hadamard matrix.