Is there a way (perhaps through Twitter's API / official stats, etc.) to see a daily graph of how many tweets linked to my domain?

Perhaps more of a DIY solution, but I'll mention it anyway.

Logstash has a twitter plugin. You can visualize it with Kibana. It should all be open source and free. There's an ELK stack (Elastic, Logstash, Kibana) docker image that has all of them combined.

YouTube: Twitter analysis using ELK

Edit: this seems a free analytics site/app but I've never tried it: tweetchup

  • Insights on tweets containing specific keywords/hashtag/user mention.
  • Stats on users mentioned keywords.
  • Interactive map with locations of users mentioned keywords.
  • Hashtags most used within tweets containing keywords.
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