Leader election is a critical and extensively studied problem in distributed computing. This paper introduces the study of leader election using mobile agents. Consider n agents initially placed arbitrarily on the nodes of an arbitrary, n-node, m-edge graph G. These agents move autonomously across the nodes of G and elect one agent as the leader such that the leader is aware of its status as the leader, and the other agents know they are not the leader. The goal is to minimize both time and memory usage. We study the leader election problem in a synchronous setting where each agent performs operations simultaneously with the others, allowing us to measure time complexity in terms of rounds. We assume that the agents have prior knowledge of the number of nodes n and the maximum degree of the graph \(\varDelta \) . We first elect a leader deterministically in \(O(n \log ^2 n + D \varDelta \log n)\) rounds with each agent using \(O(\log n)\) bits of memory, where D is the diameter of the graph. Leveraging this leader election result, we then present a deterministic algorithm for constructing a minimum spanning tree of G in \(O(m + n \log n)\) rounds, with each agent using \(O(\varDelta \log n)\) bits of memory. Finally, using the same leader election result, we improve time and memory bounds for other key distributed graph problems, including gathering, maximal independent set, and minimal dominating set. For all the aforementioned problems, our algorithms remain memory-optimal.