When I run a query such as match (n) return n all of the nodes are returned, however, the relationships and how the nodes are connected does not appear to be in the response dat
[EDITED]
The neo4j browser has an "autoComplete" setting (labelled "Connect result nodes" in the latest version) that enables/disables querying for the relationships between the nodes in every result set. (Click on the gear icon in the leftmost panel of the browser to see the settings.)
In other words, the neo4j browser has to perform extra queries to get those relationships -- they are not returned for free when a query just asks for nodes.
If you are curious how autoComplete is implemented, here is a link to some of the actual code.