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I am interested in learning about differential graph theory or differential operators on graphs, something related to what E. Bautista introduced in his answer here. 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.)

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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

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.
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I would suggest to look at the book listed below. I guess you can find it on the internet.

P. M. Soardi, Potential theory on infinite networks. Lecture Notes in Math., 1590 Springer-Verlag, Berlin, 1994.

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