  Leader election is one of the fundamental and well-studied problems in
distributed computing. In this paper, we initiate the study of leader election
using mobile agents. Suppose $n$ agents are positioned initially arbitrarily on
the nodes of an arbitrary, anonymous, $n$-node, $m$-edge graph $G$. The agents
relocate themselves autonomously on the nodes of $G$ and elect an agent as a
leader such that the leader agent knows it is a leader and the other agents
know they are not leaders. The objective is to minimize time and memory
requirements. Following the literature, we consider the synchronous setting in
which each agent performs its operations synchronously with others and hence
the time complexity can be measured in rounds. The quest in this paper is to
provide solutions without agents knowing any graph parameter, such as $n$, a
priori. We first establish that, without agents knowing any graph parameter a
priori, there exists a deterministic algorithm to elect an agent as a leader in
$O(m)$ rounds with $O(n\log n)$ bits at each agent. Using this leader election
result, we develop a deterministic algorithm for agents to construct a minimum
spanning tree of $G$ in $O(m+n\log n)$ rounds using $O(n \log n)$ bits memory
at each agent, without agents knowing any graph parameter a priori. Finally,
using the same leader election result, we provide improved time/memory results
for other fundamental distributed graph problems, namely, gathering, maximal
independent set, and minimal dominating sets, removing the assumptions on
agents knowing graph parameters a priori.
