The large dataset of charm decays collected by $\mbox{LHCb}$ in the first data-taking period ($\text{Run}~1$) of the Large Hadron Collider ($\mbox{LHC}$), facilitates precision measurements of charm mixing and of the Charge-Parity (${CP}$) symmetry. Of particular interest are two-body decays due to their large branching fractions and the abundance of Cabibbo favoured (CF) control modes, suppressed modes and ${CP}$ eigenstates. The observable $A_{\Gamma}$, which is predominantly a measure of indirect ${CP}$ violation, is accessed through the singly Cabibbo suppressed decays $\mathrm{D}^{0} \rightarrow \mathrm{K}^{+} \mathrm{K}^{-}$ and $\mathrm{D}^{0} \rightarrow \pi^{+} \pi^{-}$. The results are $A_{\Gamma}(\mathrm{K}^{+} \mathrm{K}^{-}) = (-0.30 \pm 0.32 \pm 0.10)\times 10^{-3}$ and $A_{\Gamma}(\pi^{+} \pi^{-}) = (0.46 \pm 0.58 \pm 0.12)\times 10^{-3}$, where the first uncertainty is statistical, and the second systematic. These measurements show no evidence of ${CP}$ violation and improve on the precision of the previous best measurements by nearly a factor of two. The interference between possible decay mechanisms of the doubly Cabibbo suppressed (DCS) $\mathrm{D}^{0} \rightarrow \mathrm{K}^{+}\pi^{-}$ and ${\kern 0.2em\overline{\kern -0.2em \mathrm{D}}{}}{}^{0} \rightarrow \mathrm{K}^{-}\pi^{+}$ decays gives access to the charm system mixing parameters and provides a test of various ${CP}$ violation hypotheses. The results are consistent with the no ${CP}$ violation hypothesis.