This paper presents methods to find a positively invariant set (PIS) and derive conditions on the edge weights for a multirate Kuramoto oscillator network to achieve synchronization. These methods can be applied to a network with an arbitrary topology and nonidentical oscillators. The proposed methods are based on the construction of energy functions for this type of network. Two different conditions on the edge weights are provided by using graph spectral properties, or alternatively by analyzing a path set of a graph. These methods provide flexibility in checking the edge weight conditions. We further improve the estimate of the PIS for the multirate Kuramoto network when its damping coefficients are greater than a certain value. The effectiveness and conservativeness of the proposed methods are demonstrated by simulation studies. A comparison with another method from the existing literature shows that our method gives less conservative estimate of the PIS.