  This study investigates the strain-induced structural transitions of $\eta
\leftrightarrow \theta$ and the changes in electronic band structures of
Au$_2$X (X=S, Se, Te, Si, Ge) and Au$_4$SSe. We focus on Au$_2$S monolayers,
which can form multiple meta-stable monolayers theoretically, including
$\eta$-Au$_2$S, a buckled penta-monolayer composed of a square Au lattice and S
adatoms. The $\theta$-Au$_2$S is regarded as a distorted structure of
$\eta$-Au$_2$S. Based on density functional theory (DFT) calculations using a
generalized gradient approximation, the conduction and the valence bands of
$\theta$-Au$_2$S intersect at the $\Gamma$ point, leading to linear dispersion,
whereas $\eta$-Au$_2$S has a band gap of 1.02 eV. The conduction band minimum
depends on the specific Au-Au bond distance, while the valence band maximum
depends on both Au-S and Au-Au interactions. The band gap undergoes significant
changes during the $\eta \leftrightarrow \theta$ phase transition of Au$_2$S
induced by applying tensile or compressive in-plane biaxial strain to the
lattice. Moreover, substituting S atoms with other elements alters the
electronic band structures, resulting in a variety of physical properties
without disrupting the fundamental Au lattice network. Therefore, the family of
Au$_2$X monolayers holds potential as materials for atomic scale network
devices.
