  In this paper, we investigate the leader election problem in diameter-two
networks. Recently, Chatterjee et al. [DC 2020] studied the leader election in
diameter-two networks. They presented a $O(\log n)$-round deterministic
{implicit} leader election algorithm which incurs optimal $O(n\log n)$
messages, but a drawback of their algorithm is that it requires knowledge of
$n$. An important question -- whether it is possible to remove the assumption
on the knowledge of $n$ was left open in their paper. Another interesting open
question raised in their paper is whether {\em explicit} leader election can be
solved in $\tilde{O}(n)$ messages deterministically. In this paper, we give an
affirmative answer to them. Further, we solve the {\em broadcast problem},
another fundamental problem in distributed computing, deterministically in
diameter-two networks with $\tilde{O}(n)$ messages and $\tilde{O}(1)$ rounds
without the knowledge of $n$. In fact, we address all the open questions raised
by Chatterjee et al. for the deterministic leader election problem in
diameter-two networks. To the best of our knowledge, this is the first
$\tilde{O}(n)$ deterministic result for the explicit leader election in the
diameter-two networks, that too without the knowledge of $n$.
