  Graph theory is important in information theory. We introduce a quantization
process on graphs and apply the quantized graphs in quantum information. The
quon language provides a mathematical theory to study such quantized graphs in
a general framework. We give a new method to construct graphical quantum error
correcting codes on quantized graphs and characterize all optimal ones. We
establish a further connection to geometric group theory and construct quantum
low-density parity-check stabilizer codes on the Cayley graphs of groups. Their
logical qubits can be encoded by the ground states of newly constructed exactly
solvable models with translation-invariant local Hamiltonians. Moreover, the
Hamiltonian is gapped in the large limit when the underlying group is infinite.
