*Disclaimer*: This question [was initially](https://math.stackexchange.com/q/4718589) asked yesterday in Mathematics Stack Exchange but left unanswered there.




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I am interested in learning about differential graph theory or differential operators on graphs, something related to what [E. Bautista](https://mathoverflow.net/users/173461/e-bautista) introduced in his answer [here](https://mathoverflow.net/questions/392391/differentiation-of-functions-over-graphs). Can one suggest a textbook in which such topics are discussed? (The quoted answer refers to some papers, but I prefer a book covering a comprehensive treatment of the topic.)



Reference request for differential graph theory
Well, here's a recent monograph on different flavours of graph Laplacians by Kostenko and Nicolussi: [https://www.mat.univie.ac.at/~kostenko/list/GraphLaplInf.pdf](https://www.mat.univie.ac.at/%7Ekostenko/list/GraphLaplInf.pdf)


But I'd say that the topic is too vast to be sufficiently covered in any single coherent book; you have such different appearances of operators related to edgewise derivative! To list just a few:


* random walks on finitely generated groups (there are thousands of papers on this!)
* expander graphs, and other isoperimetric properties related to amenability (Cheeger constants, cost of a group)
* geodesic flow on hyperbolic groups
* spectral theory of random graphs
* Selberg-like trace formulas
* Gromov's proof of Stallings splitting theorem via harmonic functions
* *...here someone more knowledgeable may insert ten times more topics, because I do not know much of it outside of applications to group theory; I would be interested myself to see an operator which is* not *a variant of Laplacian used to do something meaningful*.


