From Combinatorics Wiki
Table of the orders of the largest known regular planar graphs for the undirected degree diameter problem
d\k  2  3  4  5  6  7  8

3  6  12  18  28  36  52  76

4  9  16  27  44  81  134  243

5  NA  16  28  62  124  254  500

Optimal graphs are marked in bold. The following table is the key to the colors in the table presented above:
Color  Details

*  Graphs found by Preen.

*  Graphs found by Pratt and Friedman.

Table of the orders of the largest known planar graphs for the undirected degree diameter problem
d\k  2  3  4  5  6  7  8  9  10

3  7  12  18  28  38  53  77  109  157

4  9  16  27  44  81  134  243  404  728

5  10  19  39  73  158  289  638  1153  2558

6  11  24  55  117  280  579  1405  2889  7030

7  12  28  74  165  452  984  2720  5898  16328

8  13  33  97  228  685  1590  4901  11124  33613

9  14  37  122  293  986  2338  7898  18698  63194

10  16  42  151  375  1366  3369  12301  30315  110716

Optimal graphs are marked in bold. The following table is the key to the colors in the table presented above:
Color  Details

*  Graphs of unknown author.

*  Graphs found by Fellows, Hell, and Seyffarth. Details are available in a paper by the authors.

*  Graphs found by Yang, Lin, and Dai. Details are available in a paper by the authors.

*  Graphs found by Geoffrey Exoo.

*  Graphs found by S. A. Tishchenko. Details are available in a paper by the author.

*  Graphs found by R. FeriaPurón and G. PinedaVillavicencio. Details are available in a paper by the authors.

References
 Fellows, M.; Hell, P.; Seyffarth, K. (1998), "Constructions of large planar networks with given degree and diameter", Networks 32: 275281.
 FeriaPurón, R.; PinedaVillavicencio, G. (2013), "Constructions of large graphs on surfaces", preprint, PDF version.
 Tishchenko, S. A. (2012), "Maximum size of a planar graph with given degree and even diameter", European Journal of Combinatorics 33: 380396.
 Yang, Y.; Lin, J.; Dai, Y. (2002), "Largest planar graphs and largest maximal planar graphs of diameter two", Journal of Computational and Applied Mathematics 144:349358.
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