Geodesic
The largest graphs of diameter $2$ and fixed Euler characteristics
Fundamentalʹnaâ i prikladnaâ matematika, Tome 7 (2001) no. 4, pp. 1203-1225.

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We compute the exact maximum size of a planar graph with diameter 2 and fixed maximum degree $\Delta\leq7$. To find the solution of the problem we use the irrelevant path method. It is proved that the known graphs with size $2\Delta+1$ ($3\leq\Delta\leq4$) and $\Delta+5$ ($5\leq\Delta\leq7$) are the largest possible ones. This result completes the analysis of the degree–diameter problem for planar graphs of diameter 2. In the case $\Delta\leq6$, we found also the largest graphs of diameter 2 that are embedded into the projective plane and into the torus. 
@article{FPM_2001_7_4_a13,
     author = {S. A. Tishchenko},
     title = {The largest graphs of diameter $2$ and fixed {Euler} characteristics},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {1203--1225},
     publisher = {mathdoc},
     volume = {7},
     number = {4},
     year = {2001},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_2001_7_4_a13/}
}
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S. A. Tishchenko. The largest graphs of diameter $2$ and fixed Euler characteristics. Fundamentalʹnaâ i prikladnaâ matematika, Tome 7 (2001) no. 4, pp. 1203-1225. http://geodesic.mathdoc.fr/item/FPM_2001_7_4_a13/