There are a few different ways to open an existing graph:
Navigate to File -> Open Graph to open the File Explorer from which you can choose the graph to open. Graph Files are identified by the file suffix ".star". Alternatively, navigate to File -> Open Recent to open a recent graph without having to go through the File Explorer.
The Welcome Page also provides ways to open a graph. One option is clicking on the Open button which will open up the File Explorer. Each recent graph also has its own button (up to 10 recent graphs) which when clicked will open up the corresponding graph.
You can also drag and drop a graph into Constellation. Choose the graph file you want to open and drag it into the tab area at the top of the graph window (you will see a "+Copy" when you can drop it).