Recommendations for a book on Graph Theory?

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I am currently taking a combinatorics class that surveys a little bit of graph theory and it piqued my interest. Does anyone have a recommendation for a good introductory book on the subject? I am really interested in finding a book that is very readable and not the standard definition, lemma, theorem, proof, etc. style. I don't know if that is common for texts in the fields of combinatorics and graph theory or not. I am also very interested in applications, however I am not a computer science student and I don't know if that's all an introductory book could be applied to. If at all possible, I would love to find a book that had applications to physics.

Any help would be greatly appreciated!
 
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Thank you very much!
 
Oh wow, that's cool. Thanks for the recommendation and trivia!