Please have a look at my paper at Academia.edu because it did not load right here and I put it there. It is basically a Circulant Graph with a Star that connects to all the vertices of the Circulant from inside of it. It is supposed to be a solution to describe human networks in certain situations.
https://www.academia.edu/s/9a2552ae28 is still converting to their format. If the graphs still do not look good, please write to me and ask to have a copy of this draft. I will e-mail you one for free for your perusal. You will also find a copy of this paper attached to this message.
https://www.academia.edu/s/9a2552ae28