She struggled to settle into reception last year,which I believed was because she didn't know anyone before school (she'd been full time at a private nursery up to that point). She seemed to spend a lot of time on the 'friendship bench', and with a buddy from an older year. She also spent a lot of time chatting with the lunchtime supervisors.
She gradually seemed to settle and now has 2 friends. I believed that she was getting along with all her classmates.
Another mum recently told me that last year my DD used to chat with her every lunchtime (the mum was doing some work at the school).
DD has just told me that she walks with the lunchtime supervisors most days. She asks if she can join in with her classmates' play but they say no. 4 boys were throwing grass at her today which she told them she didn't like. They didn't stop.
It breaks my heart that she's so lonely. She will soon start to notice that no one seems yo want to spend time with her.
But ... is she unhappy?? Some kids at that age (my DD for one) actually find talking to adults quite interesting and like it. Has she said that she's lonely in so many words? The fact that she's an only, IMO, is likely to mean she is comfortable talking to grown-ups.
The ratio of girls to boys in DD's year 1 class is 1:3. DD spends most of her time playing with the boys and rarely mentions the girls. She was in preschool with them since age 3, but in Reception she didnt mix too well with the girls.
You could invite a couple of girls round for a playdate, maybe one at a time would be best, in case they gang up against DD.
My feeling is that you don't want to project your worries on to your DD and let her sense social interaction is a fraught business. If she isn't actually unhappy I would take a step back and let her find her own way.
I do understand though, my DD is a bit like your DD and it's hard.
It really does help inviting people round. Build up gradually to include 3 or 4 dc coming round one at a time to give her some options when she's at school. If she's tired in the week, ask if they could come at the weekend.
But just to say there will be lots of them feeling this way at one time or another. I know mine had phases of it.
It does settle down as year 1 goes on. They're put into groups for working and often they get on with someone on their table. Friendship groups change all the time - particularly from reception to year 1.
And the boys throwing grass - there's always two or three like this - they're probably throwing grass at everyone. Try not to think it's just your dd they're targeting. It's highly unlikely that they would.
Activities out of school are good too. Sometimes you'll find another from her class there and they tend to cling together. If not, it gives them confidence to make a friend outside school. My dd has made a number of friends at gymnastics and she's very shy.