  A nonsplit graph is a directed graph where each pair of nodes has a common
incoming neighbor. We show that the radius of such graphs is in $O(\log \log
n)$, where $n$ is the number of nodes. We then generalize the result to
products of nonsplit graphs.
  The analysis of nonsplit graph products has direct implications in the
context of distributed systems, where processes operate in rounds and
communicate via message passing in each round: communication graphs in several
distributed systems naturally relate to nonsplit graphs and the graph product
concisely represents relaying messages in such networks. Applying our results,
we obtain improved bounds on the dynamic radius of such networks, i.e., the
maximum number of rounds until all processes have received a message from a
common process, if all processes relay messages in each round. We finally
connect the dynamic radius to lower bounds for achieving consensus in dynamic
networks.
