I am trying to display only one relevant relationship in the cypher query web browser but it displays all the relationships in between nodes. I am running following query:
I believe that these relationships are only present in the Graph Visualization Mode of Neo4j Browser. If you change your visualization mode to "Text", "Table" or "Code" these relationships will not be shown. That is: the Graph visualization mode is trying to "complete" the graph for you.
To achieve the desired result you should go to the section "Graph Visualization" of Neo4j Browser Settings and uncheck the option "Connect result nodes" as show in the image below: