How do you prepare to attend a conference where you may have trouble following the talks?
Nobody can know everything. Even just in mathematics the last time it was possible to literally know it all was early in the 20th century. Things have advanced too fast and too far for such things now.
A conference might be expected to present the latest ideas in a sub field. If that is your specialty you would be expected to follow &quot;generally&quot; but would probably require additional study afterwards if the subject of a talk was important to your work. But a &quot;general understanding&quot; and a few notes about how and where to follow up is probably enough for most people.
I doubt that your education, as you describe it, puts you at the research edge in graph theory. If you are doing research in that, then a quick literature survey of closely relevant topics would be good preparation, but I'd guess that few  people do that. If the proceedings are available in advance, skimming the papers will give you an idea of where the holes are in your understanding.
But at the conference take a few notes, especially on topics that are not known to you, as a guide for further study. If you attend (physically or virtually) with colleagues, start a conversation over coffee/tea/whatever to ask others for their insights.
On the other hand, if one of the speakers is your dissertation advisor, you'd probably be expected to understand most of it. Or at least grasp the general trend of thought.
Math is pretty esoteric at the edges.

Another kind of &quot;preparation&quot;, however, is to prepare to meet new people and share a few ideas. Harder now with thing mostly virtual, but a great way to build contacts.