Moore Graph

A Graph with Diameter $d$ and Girth $2d+1$. Moore graphs have Diameter of at most 2. Every Moore graph is both Regular and distance regular. Hoffman and Singleton (1960) show that $k$-regular Moore graphs with Diameter 2 have $k\in \{2, 3, 7, 57\}$.

References

Godsil, C. D. ``Problems in Algebraic Combinatorics.'' Electronic J. Combinatorics 2, F1 1-20, 1995. http://www.combinatorics.org/Volume_2/volume2.html#F1.

Hoffman, A. J. and Singleton, R. R. ``On Moore Graphs of Diameter Two and Three.'' IBM J. Res. Develop. 4, 497-504, 1960.