<p>In this article, we introduce new models and algorithms that extend the classical network flow problems to the setting with electric vehicles (EV) that accommodate EV-specific constraints such as range limitations, charging strategies, and station capacities. Our work focuses on solving three key problems: single EV optimal charging strategy, maximum EV flow, and minimum-cost EV flow, each central to the efficient operation of EV routing systems. We establish the computational complexity of these problems, demonstrating their NP-hardness in general settings, while also identifying precise conditions under which they become polynomial-time solvable. For these tractable cases, we develop exact algorithms, and for the general settings, we design fully polynomial-time approximation schemes (FPTAS). We conduct numerical experiments using a network calibrated with real-world data. Although the conditions for polynomial time solvability do not hold in this setting, our algorithm still computes the optimal solution, which demonstrates its scalability and practical relevance.</p>