This would depend a bit on your background. If you are familiar with fuzzy sets, then <A HREF="https://www.springer.com/gp/book/9783319714066">Fuzzy Graph Theory</A> by Mathew, Mordeson, and Malik offers a good overview. They also have a <A HREF="https://www.springer.com/gp/book/9783319764535">follow-up volume</A> with more specialized applications.

If you are not familiar with fuzzy sets, you could start with <A HREF="https://www.semanticscholar.org/paper/Introduction-to-Fuzzy-Graph-Theory-Pal/ed409d98e11d0732ef21e92a959fc6a657d1ac56">Introduction to Fuzzy Graph Theory</A>, just to get a feeling for the field, but then you would want to first study fuzzy sets, perhaps by following this <A HREF="https://www.youtube.com/watch?v=IZWTduVCrf8">online course.</A> (I have not found an online course specifically on fuzzy graphs.)