  We model monolayer graphene-like materials with BC$_6$N stoichiometry where
the bonding between the B and the N atoms plays an important role for their
physical and chemical properties. Two types of BC$_6$N are found based on the
BN bonds: In the presence of BN bonds, an even number of $\pi$-bonds emerges
indicating an aromatic structure and a large direct bandgap appears, while in
the absence of BN bonds, an anti-aromatic structure with an odd-number of
$\pi$-bonds is found resulting a direct small bandgap. The stress-strain curves
shows high elastic moduli and tensile strength of the structures with BN-bonds,
compared to structures without BN-bonds. Self-consistent field calculations
demonstrate that BC$_6$N with BN-bonds is energetically more stable than
structures without BN-bonds due to a strong binding energy between the B and
the N atoms, while their phonon dispersion displays that BC$_6$N without
BN-bonds has more dynamical stability. Furthermore, all the BC$_6$N structures
considered show a large absorption of electromagnetic radiation with
polarization parallel to the monolayers in the visible range. Finer detail of
the absorption depend on the actual structures of the layers. A higher
electronic thermal conductivity and specific heat are seen in BC$_6$N systems
caused by hot carrier--assisted charge transport. This opens up a possible
optimization for bolometric applications of graphene based material devices.
