On spectra of doubly regular asymmetric digraphs of \(RH\)-type (Q1812788)

Let \(v\), \(\kappa\), \(\lambda\) be positive integers. A digraph \({\mathcal D}=(V,U)\) of order \(v\) is a doubly regular asymmetric \((v,\kappa,\lambda)\) digraph if (i) \(\mathcal D\) is regular of valency \(k\), (ii) \(\mathcal D\) is doubly regular of double valency \(\lambda\), and (iii) \(\mathcal D\) is asymmetric. \(\mathcal D\) is of regular Hadamard (RH) type if furthermore \(v=4(k-\lambda)\). Associated with \(\mathcal D\) is a graph \(N(\mathcal D)=(V,E)\), where \(\{a,b\}\in E\) if and only if \(a\) and \(b\) are not adjacent in \(\mathcal D\). Let \(A\) be the adjacency matrix of \(\mathcal D\) and \(N\) the adjacency matrix of \(N(\mathcal D)\). The author investigates the spectra of \(A\) and \(N\) when \(\mathcal D\) is of RH-type.