Why has this been submitted, I mean I've seen few versions of "calculus on graphs", is it because of the "translation" of analysis to graph theory? Could the submitter explain their interest in this and why they think its relevant?

The downvotes on this post are a shame. I don't think every post should have to be defended for its relevance (especially since the explicit mission of the site is to gather news of possible interest to hackers, not just hacking news), but it seems like a reasonable question to what the submitter thinks an interested non-expert should direct his or her attention.

I like to refer to DNNs (specifically back-propagation on them) as the world's most lucrative application of the chain rule from Calculus...

Somewhat related, but my old PhD supervisor had a very interesting paper on extending some ideas from Riemannian geometry to graphs. http://arxiv.org/abs/1011.5898

I spent a summer as an undergrad doing research on this sort of thing, a nice paper came out of it from the grad students working with us [0].

I love to see graph theory creeping into other branches of mathematics, especially blending the discrete and continuous domains.

0 - https://books.google.com/books?id=J65sAwAAQBAJ&pg=PA15&lpg=P...

Standalone version here: http://people.math.gatech.edu/~mbaker/pdf/Quantum_Graphs_Rev...