  We study the communication complexity of a number of graph properties where
the edges of the graph $G$ are distributed between Alice and Bob (i.e., each
receives some of the edges as input). Our main results are:
  * An Omega(n) lower bound on the quantum communication complexity of deciding
whether an n-vertex graph G is connected, nearly matching the trivial classical
upper bound of O(n log n) bits of communication.
  * A deterministic upper bound of O(n^{3/2}log n) bits for deciding if a
bipartite graph contains a perfect matching, and a quantum lower bound of
Omega(n) for this problem.
  * A Theta(n^2) bound for the randomized communication complexity of deciding
if a graph has an Eulerian tour, and a Theta(n^{3/2}) bound for the quantum
communication complexity of this problem.
  The first two quantum lower bounds are obtained by exhibiting a reduction
from the n-bit Inner Product problem to these graph problems, which solves an
open question of Babai, Frankl and Simon. The third quantum lower bound comes
from recent results about the quantum communication complexity of composed
functions. We also obtain essentially tight bounds for the quantum
communication complexity of a few other problems, such as deciding if G is
triangle-free, or if G is bipartite, as well as computing the determinant of a
distributed matrix.
