In this paper we develop an alternative topological data analysis (TDA) approach for studying graph representations of time series of dynamical systems. Specifically, we show how persistent homology, a tool from TDA, can be used to yield a compressed, multi-scale representation of the graph that can distinguish between dynamic states such as periodic and chaotic behavior. We show the approach for two graph constructions obtained from the time series. In the first approach the time series is embedded into a point cloud which is then used to construct an undirected k-nearest-neighbor graph. The second construct relies on the recently developed ordinal partition framework. In either case, a pairwise distance matrix is then calculated using the shortest path between the graph's nodes, and this matrix is utilized to define a filtration of a simplicial complex that enables tracking the changes in homology classes over the course of the filtration. These changes are summarized in a persistence diagram's two-dimensional summary of changes in the topological features. We then extract existing as well as new geometric and entropy point summaries from the persistence diagram and compare to other commonly used network characteristics. Our results show that persistence-based point summaries yield a clearer distinction of the dynamic behavior and are more robust to noise than existing graph-based scores, especially when combined with ordinal graphs.