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 $\omega$-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 $\alpha$ 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 $\alpha$-exponential matrix and illustrate the solutions through numerical simulation in various examples.