On graphs with diameter 2
Consider the set of those graphs with p points in which whenever two points are not joined by a line then there are at least k points joined to both of them. We prove that if p≥2k then among these graphs there is exactly one with minimal number of lines, the complete graph with k points in one class. This extends a result of Murty and Vijayam. © 1976 Academic Press, Inc.
Journal of Combinatorial Theory, Series B
Bollobás, B., & Eldridge, S. (1976). On graphs with diameter 2. Journal of Combinatorial Theory, Series B, 21 (3), 201-205. https://doi.org/10.1016/S0095-8956(76)80003-2