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Let $\mathcal{G}_{n,p,q}$ denote the set of strongly connected bipartitedigraphs on $n$ vertices which contain a complete bipartite subdigraph$\overleftrightarrow{K_{p,q}}$, where $p, q, n$ are positive integers and $p+q\leq n$. In this paper, we study the Q-index (i.e. the signless Laplacianspectral radius) of $\mathcal{G}_{n,p,q}$, and determine the extremal digraphthat has the minimum Q-index.

The minimum Q-index of strongly connected bipartite digraphs with complete bipartite subdigraphs