Weakly Hadamard diagonalizable graphs

The EKR-Module Property of Pseudo-Paley Graphs of Square Order

“…C. H. Yip is supported by a Four Year Doctoral Fellowship from the University of British Columbia. C. H. Yip thanks Shaun Fallat for the clarification of a theorem in [1]. The authors are also grateful to the referees for valuable comments, corrections, and suggestions.…”

“…If q ≡ 1 (mod 4), then the Paley graph P q is a strongly regular graph with parameters (q, q−1 2 , q−5 4 , q−1 4 ). Moreover, the eigenvalues of P q are q−1 2 (with multiplicity 1), 1 2 (−1 ± √ q) (each with multiplicity q−1 2 ). Weng, Qiu, Wang, and Xiang [33] introduced the definition of pseudo-Paley graphs.…”

“…A graph Γ is called Hadamard diagonalizable if the Laplacian of Γ can be diagonalized by a Hadamard matrix [6]. Recently, Adm et al [1] studied a larger class of graphs which contains some families of strongly regular graphs. In order to present this result, they introduced a broader class of matrices which include Hadamard matrices.…”

“…Paley graphs are known to have a close connection with Paley's construction on Hadamard matrices [26]. Recently, Adm et al [1] introduced the notion of weakly Hadamard matrices and studied weakly Hadamard diagonalizable graphs (see Definition 19 and 20). In particular, they showed that Paley graphs of square order are weakly Hadamard diagonalizable [1, Theorem 5.9].…”