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The phase space for graphene's minimum conductivity $\sigma_\mathrm{min}$ ismapped out using Landauer theory modified for scattering using Fermi's GoldenRule, as well as the Non-Equilibrium Green's Function (NEGF) simulation with aMonte Carlo sampling over impurity distributions. The resulting `fan diagram'spans the range from ballistic to diffusive over varying aspect ratios ($W/L$),and bears several surprises. {The device aspect ratio determines how muchtunneling (between contacts) is allowed and becomes the dominant factor for theevolution of $\sigma_{min}$ from ballistic to diffusive regime. We find anincreasing (for $W/L>1$) or decreasing ($W/L<1$) trend in $\sigma_{min}$ vs.impurity density, all converging around $128q^2/\pi^3h\sim 4q^2/h$ at the dirtylimit}. In the diffusive limit, the {conductivity} quasi-saturates due to theprecise cancellation between the increase in conducting modes from chargepuddles vs the reduction in average transmission from scattering at the DiracPoint. In the clean ballistic limit, the calculated conductivity of the lowestmode shows a surprising absence of Fabry-P\'{e}rot oscillations, unlike othermaterials including bilayer graphene. We argue that the lack of oscillationseven at low temperature is a signature of Klein tunneling.

Quantum transport at the Dirac point: Mapping out the minimum conductivity from pristine to disordered graphene