ok so here is the only real data I could find. At the end of the paper:
"The LSG-EC can exhibit energy densities of up to 1.36 mWh/cm3 , a value that is approximately two times higher than that of the AC-EC"
1.36 mWh/cm3 = 1.36 Wh/L
If we assume that 1 L of it weights very roughly 1kg.
We get 1.36 Wh/kg.
From wikipedia:
"The amount of energy stored per unit weight [in ultracapacitors] is generally lower than that of electrochemical batteries (3 to 5 W·h/kg, although 85 W·h/kg has been achieved in the lab[12] as of 2010 compared to 30 to 40 W·h/kg for a lead acid battery, 100 to 250 W·h/kg for a lithium-ion battery and about 0.1% of the volumetric energy density of gasoline."
This is still less than 1% of the energy density of lithium ion batteries.
There is a big miscalculation: Graphene seems to be extremely light weight. 1 liter of graphene will definitely not weight 1kg. Wikipedia [1] only gives measures for m²: Graphene is very light, weighing only about 0.77 milligrams per square meter.
Here[1], the density is said to be as low as 0.03 gm/cc which is the same as kg/liter. That gives a multiplicative factor of 33 over the above calculations, i.e. this could be 33% of the energy density of lithium ion batteries. Not bad.
Energy per liter also is an important factor. Your mobile would float, but it also would be the size of your lower arm. Maybe, you should wear your battery as a jacket or a pair of trousers?
A square meter of a single-atom-thick layer of graphene isn't very relevant. Expect the final product to have a volumetric density in the neighborhood of graphite or diamond.
Graphene is also very thin. If you created a box of graphene, it would naturally have about the same weight as diamond or coal (plus or minus 100%, maybe, but certainly not orders of magnitude less).
"The LSG-EC can exhibit energy densities of up to 1.36 mWh/cm3 , a value that is approximately two times higher than that of the AC-EC"
1.36 mWh/cm3 = 1.36 Wh/L
If we assume that 1 L of it weights very roughly 1kg. We get 1.36 Wh/kg.
From wikipedia:
"The amount of energy stored per unit weight [in ultracapacitors] is generally lower than that of electrochemical batteries (3 to 5 W·h/kg, although 85 W·h/kg has been achieved in the lab[12] as of 2010 compared to 30 to 40 W·h/kg for a lead acid battery, 100 to 250 W·h/kg for a lithium-ion battery and about 0.1% of the volumetric energy density of gasoline."
This is still less than 1% of the energy density of lithium ion batteries.