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This matrix can be block-partitioned into \(q \times q\) blocks, with$$\begin{aligned} (M_q)_{ij} = \left\{ \begin{array}{ll} q I_q - \mathbf{1}_{q \times q} , &{} \text{ if } j-i \equiv 0 ~\mathrm{mod}~q , \\ 0, &{} \text{ otherwise } . \end{array}\right. \end{aligned}$$ Now consider the subcase that \(l \equiv 0 ~\mathrm{mod}~q\). Then \(B^\top B\) consists of l/q repetitions of \(M_q\) on each block row and block column of size \(q^2 \times q^2\). Therefore each such row and column block gives \((q-1)q\) eigenvalues of magnitude l and rest of the eigenvalues will be zero. Next, if \(l ~\mathrm{mod}~q =: k \ne 0\), then \(B^\top B\) consists of \((l-k)/q\) repetitions of \(M_q\) on each block row and column of size \(q^2 \times q^2\). In addition it contains first k column blocks of \(M_q\) concatenated l times column-wise as the last block of columns. It contains first k row blocks of \(M_q\) concatenated \(l-k\) times row-wise as the last block of rows. The extra k rows and columns of \(B^\top B\) give \((q-1)k\) eigenvalues of magnitude \(l+(q-k)\). Another set of row and column blocks gives \((q-1)(q-k)\) eigenvalues of magnitude \(l-k\). The remaining eigenvalues are 0. \(\square \) Remark It has been pointed out to the authors by the referee that the construction in Theorem 3 can be viewed as an explicit realization of the results of [20] (in particular [20, Corollary 2.2]). For any positive integers m and n, the complete bipartite graph \(K_{m,n}\) is immediately seen to be a Ramanujan bigraph as its eigenvalues are \(\{\pm \sqrt{mn},\,0\}\). The Ramanujan bigraphs constructed in Items (1) and (2) of Theorem 3 are explicit constructions of q-coverings of \(K_{l,q}\) and \(K_{q,l}\), respectively. We also note that when \(l = q\), Theorem 3 leads to the following corollary. Corollary 1 For every prime number q, the matrix B(q, q) defined in (6) is the biadjacency matrix of a \((q^2,q^2)\)-bipartite Ramanujan graph with degree q. In this case \(\lambda _1 = q\), and \(\lambda _2 = \sqrt{q}\) with a multiplicity of \(q(q-1)\). The remaining \(q-1\) eigenvalues of B(q, q) are zero. This leads to a new class of \((q^2,q^2)\)-bipartite