I published a paper, the code for which is uploaded in GitHub.
Is it possible to add the GitHub link as an entry in Google Scholar? If so, can the citations be monitored in the same way as regular publications?
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The way with dealing with github repositories in a citable way is to draw a DOI from Zenodo. Zenodo allows you to create a tag or release in the github repository and make that citable.
I don't know specifically whether google scholar indexes that, but at least it is something you can put into a list of references.
There are two aspects:
Instead, you might want to give people a hint to cite the associated paper. To this end, add a disclaimer at a prominent location in your repository's Readme.MD file:
Cite as: (bibliographic information for the associated paper)
Google indexes things by crawling the internet, you don't usually add things to Google except to help collect the things Google has found into your personal profile (you can also manually add items to your profile that Google has not indexed, but I do not believe these will appear in searches by others or have any other functionality besides appearing in your profile; see also https://academia.stackexchange.com/a/143964/63475).
Generally, if there is a paper that goes with some code, people will cite the paper rather than github. If they link to github this will probably be in their code, not their paper, and isn't an academic citation but rather attribution as often required by the license you distribute your work under. There are even particular journals like https://www.jstatsoft.org/index that primarily publish papers that describe/link to software packages.
You can cite the URLs of your repository with @misc in your BibTeX file (see https://tex.stackexchange.com/questions/35977/how-to-add-a-url-to-a-latex-bibtex-file )
If your goal is to receive indexed citations, the clean way to do it is to write in the beginning of your README something like:
"This repository contains the code relative to [paper]. Please cite [paper] when referring to this repository."