How do I search for the emails that belong only to a specified label, and do not belong to any other labels.
You can search for emails that belong to a specific label using the
label:mylabel <include search terms here>
syntax, and you can exclude messages from a specific label using the
-label:mylabel <include search terms here>
So what you want can be accomplished using a combination of these both. You have to compose a search string containing a “-label:mylabel” for all the labels that you have defined, except for the one you are looking for, like this:
label:thisisthelabelimlookingfor -label:label1 -label:label2 ... <include search terms here>