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

## TL;DR

This paper investigates the minimum signless Laplacian spectral radius (Q-index) among strongly connected bipartite digraphs containing a complete bipartite subdigraph, identifying the extremal graph with the smallest Q-index.

## Contribution

It determines the extremal strongly connected bipartite digraph with the minimum Q-index within a specified class containing a complete bipartite subdigraph.

## Key findings

- Identifies the extremal digraph with the minimum Q-index.
- Provides bounds and characterization for the Q-index in this class.
- Enhances understanding of spectral properties of bipartite digraphs.

## Abstract

Let $\mathcal{G}_{n,p,q}$ denote the set of strongly connected bipartite digraphs 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 Laplacian spectral radius) of $\mathcal{G}_{n,p,q}$, and determine the extremal digraph that has the minimum Q-index.