simple lower bound for constructing a Spanning tree
i tried to solve this problem in this way:

Assume that there exists a correct SPT protocol $A$ thatuses fewer than $m$ messages. This means that there is at least one link in $G$ where no message is transmitted.
Consider an execution of the algorithm on $G$, and let $e = (x, y) \in E$ be
the link where no message is transmitted by $A$.

Now construct a new graph $G^{'}$  from $G$
by removing the edge $e$ and adding a new node $z$ and two new edges $e1 = (x, z)$ and
$e2 = (y, z)$. Set $z$ in a noninitiator status ($z$ does not starts the execution of the algorithm). Run exactly the same execution
of $A$ on the new graph $G$   : since no message was sent along $(x,y)$ in the original execution in $G$, $x$ and $y$ never send a message to $z$ in the current execution in $G^{'}$   ; and since $z$ is not an initiator and does not receive any message, it will not send any message. 

At the end of the alghoritm a tree $T$ should have been constructed,
however, $z$ is not part of $T$, and hence $T$ does not span $G$.

in other words $A$ works only if $z$  is in a initiator status, therefore $A$ is not correct.