We study persistent query evaluation over streaming graphs, which is becoming increasingly important. We focus on navigational queries that determine if there exists a path between two entities that satisfies a user-specified constraint. We adopt the Regular Path Query (RPQ) model that specifies navigational patterns with labeled constraints. We propose deterministic algorithms to efficiently evaluate persistent RPQs under both arbitrary and simple path semantics in a uniform manner. Experimental analysis on real and synthetic streaming graphs shows that the proposed algorithms can process up to tens of thousands of edges per second and efficiently answer RPQs that are commonly used in real-world workloads.