Is there a 7-regular graph on 50 vertices with girth 5? What about 57-regular on 3250 vertices?
The following problem is homework of a sort -- but homework I can't do!
The following problem is in Problem 1.F in Van Lint and Wilson:
I assigned the lower bound early on in my graph theory course. Solutions for $d=2$ and $d=3$ are easy to find. Then, last week, when I covered eigenvalue methods, I had people use them to show that there were no solutions for $d=4$, $5$, $6$, $8$, $9$ or $10$. (Problem 2 here.) I can go beyond this and show that the only possible values are
Does anyone know if the last two exist? I'd like to tell my class the complete story.