Table 2 Redundant spectra of basic symmetric motifs (BSMs) with one or two orbits. A BSM with one orbit is a uniform graph \({K}_{n}^{\alpha ,\beta }\) with n vertices and adjacency matrix \({A}_{n}^{\alpha ,\beta }=({a}_{ij})\), where a_ij = α if i ≠ j and a_ii = β, for all i, j and some constants α and β. A BSM with two orbits consists of the (γ, δ)-uniform join of two uniform graphs \({K}_{n}^{{\alpha }_{1},{\beta }_{1}}\) and \({K}_{n}^{{\alpha }_{2},{\beta }_{2}}\), that is, the graph with 2n vertices and block adjacency matrix (after a suitable labelling of the vertices) of the form \(\left(\begin{array}{ll}A & B \\ C & D\end{array}\right)\), where \(A={A}_{n}^{{\alpha }_{1},{\beta }_{1}}\), \(B={A}_{n}^{{\alpha }_{2},{\beta }_{2}}\) and \(C={A}_{n}^{\gamma ,\delta }\), each defined as above. We write e_i for the vector with non-zero entries 1 at position 1, and  − 1 at position i (2 ≤ i ≤ n), κ₁ and κ₂ for the two solutions of the quadratic equation cκ² + (−a + b)κ − c = 0, where a = α₁ − β₁, b = α₂ − β₂ and c = γ − δ, and use (v∣w) to represent the concatenation of two vectors.

From: Exploiting symmetry in network analysis