I cannot reproduce this problem.
I downloaded your file, selected the top left edge, clicked the "..." button from the edge's "Text" property, removed all the whitespace, saved the file, re-started yEd, and re-opened the file. No extra whitespace reappeared - the text was exactly as saved. I also tried all other text editing methods yEd offers and it always worked as expected.
Are you sure you are actually saving your edits? And if so, are you sure you are opening the modified file and not the original, unmodified one? Moreover, have you tried saving your edits to a new file instead of overwriting the original file?
This is the result of my edits (I changed the line color of the edited edge to red).