Normalized solutions for nonlinear Schrödinger equations on graphs

We are concerned with the nonlinear Schrödinger equation with an $L^2$ mass constraint on both finite and locally finite graphs and prove that the equation has a normalized solution by employing variational methods. We also pay attention to the behaviours of the normalized solution as the mass constraint tends to $0^{+}$ or $+\mathrm{\infty }$ and give clear descriptions of the limit equations. Finally, we provide some numerical experiments on a finite graph to illustrate our theoretical results.