existence of solutions for time fractional order diffusion equations on weighted graphs

We generalize the concept of diffusion equations on weighted graphs, which is also known as ω-diffusion equations, to study fractional order diffusion equations on weighted graphs. more precisely, we replace the ordinary first order derivative in time by a fractional derivative of order α in the sense of riemann-liouville and caputo fractional derivatives. we prove the existence of solutions of fractional order diffusion equations on graphs using the concept of α-exponential matrix and illustrate the solutions through numerical simulation in various examples.