Is it possible that as a child you had trouble making friends and that caused you to feel lonely? If that's the case you might be projecting you old feelings onto your daughter... and assuming that she's feeling what you felt. The key to the answer is in your statement "She seems happy." The person who isn't happy with your daughter's social life seems to be you.  Or maybe you are and always have been a very social person, who easily connects with others and has tons of friends. Again, you may be assuming that anyone who isn't as social as you are isn't happy. Some of us need other people to "re-charge" and some of us need to be on our own to re-charge. These are just different in-born temperaments. Neither is better than the other.

I'm glad to hear that your daughter isn't expressing loneliness to you. That's probably because she isn't lonely or sad. If your daughter gets along with friends at school and plays with the neighbor girl, and seems happy, then I don't see a problem here.

If you'd really like to get her more comfortable with the idea of inviting school friends over, how about offering your daughter the opportunity to have a sleep-over? Or a movie night? A little encouragement may be all that's needed here. If she's not keen on the idea, please don't push. Let her decide how much "friend time" she has outside of school.