Can I cite lecture notes? I’m now writing a masters thesis in mathematics. I’d like to introduce basic definitions and theorems in the beginning. By basic I mean truly basic, like the definition of a connected graph and the theorem about Eulerian paths (such a path exists if and only if even degrees). 

The lecture notes written by my professor is really nice so I’d like to use his definitions (I’ll cite, of course). In this case can I cite his lecture notes?
While there is no real harm in citing it, it is unlikely to be necessary. Common knowledge is one of the exceptions to the requirement for citation. Many things that you first see in a course fall under this cover, even if they are new to you. 

But you seem to be more speaking of copying the notes, rather than citing them. That may be a different issue if they can be considered to be copyrighted. In that case you want permission from him. 

Do a little research in a textbook or two. Do those definitions, or something essentially similar to them appear there? I suspect they do. If so, there is no need to be concerned. And also note that one of the exceptions in copyright law (most places) is that for some things there is only one effective way to state something, so it's expression can't be copyrighted. 

And it wasn't your professor who originated these ideas. 

But an acknowledgement to your professor for introducing you to these ideas would be a courteous thing to do.