A Particular Class Of Bigraphs

A semi-complete bigraph \(G\) has adjacency matrix
\[A = \begin{pmatrix} 0 & B \\ B^T & 0 \end{pmatrix},\]
where \(B + B^T = J – I\), so \(B\) is the adjacency matrix of a tournament \(T\) corresponding to \(G\). We determine algebraic and structural properties of this class of graphs. In particular, we obtain a condition equivalent to the connectedness of a semi-complete bigraph; moreover we determine characterizations of semi-complete bigraphs having 4 distinct eigenvalues in the case of \(G\) regular or \(A\) irreducible. In particular, a regular semi-complete bigraph has 4 distinct eigenvalues if and only if it corresponds to a doubly regular tournament.