I've put together a degree-diameter notebook for various graphs of the degree diameter problem. My favorite table is at The (Degree,Diameter) Problem for Graphs.
Here are sample edge lists without good known embeddings.
degreediameter62 = {{1,2},{1,3},{1,4},{1,19},{1,21},{1,30},{2,5},{2,6},{2,18},{2,24},{2,28},{3,7},{3,8},{3,20},{3,22},{3,29},{4,9},{4,10},{4,17},{4,23},{4,27},{5,10},{5,11},{5,14},{5,22},{5,26},{6,7},{6,12},{6,15},{6,25},{6,27},{7,13},{7,16},{7,23},{7,26},{8,9},{8,11},{8,14},{8,25},{8,28},{9,12},{9,15},{9,24},{9,26},{10,13},{10,16},{10,25},{10,29},{11,12},{11,20},{11,23},{11,30},{12,17},{12,21},{12,29},{13,14},{13,17},{13,24},{13,30},{14,18},{14,21},{14,27},{15,16},{15,18},{15,22},{15,30},{16,20},{16,21},{16,28},{17,19},{17,22},{17,28},{18,19},{18,23},{18,29},{19,20},{19,25},{19,26},{20,24},{20,27},{21,26},{21,31},{22,27},{22,31},{23,28},{23,31},{24,29},{24,31},{25,30},{25,31},{26,32},{27,32},{28,32},{29,32},{30,32},{31,32}};
degreediameter43 = {{1,2},{1,6},{1,26},{1,38},{2,3},{2,4},{2,5},{3,20},{3,24},{3,31},{4,13},{4,19},{4,25},{5,14},{5,18},{5,36},{6,7},{6,11},{6,35},{7,8},{7,9},{7,10},{8,25},{8,29},{8,39},{9,18},{9,24},{9,30},{10,19},{10,23},{10,31},{11,12},{11,16},{11,32},{12,13},{12,14},{12,15},{13,30},{13,34},{14,23},{14,29},{15,24},{15,28},{15,39},{16,17},{16,21},{16,38},{17,18},{17,19},{17,20},{18,33},{19,28},{20,29},{20,34},{21,22},{21,26},{21,35},{22,23},{22,24},{22,25},{23,37},{25,33},{26,27},{26,32},{27,28},{27,29},{27,30},{28,36},{30,37},{31,32},{31,41},{32,33},{33,40},{34,35},{34,41},{35,36},{36,40},{37,38},{37,41},{38,39},{39,40},{40,41}};
degreediameter53 = {{1,46},{1,52},{1,60},{1,63},{1,72},{2,47},{2,49},{2,53},{2,61},{2,64},{3,48},{3,50},{3,54},{3,62},{3,65},{4,37},{4,51},{4,55},{4,63},{4,66},{5,38},{5,52},{5,56},{5,64},{5,67},{6,39},{6,53},{6,57},{6,65},{6,68},{7,40},{7,54},{7,58},{7,66},{7,69},{8,41},{8,55},{8,59},{8,67},{8,70},{9,42},{9,56},{9,60},{9,68},{9,71},{10,43},{10,49},{10,57},{10,69},{10,72},{11,44},{11,50},{11,58},{11,61},{11,70},{12,45},{12,51},{12,59},{12,62},{12,71},{13,25},{13,33},{13,40},{13,43},{13,63},{14,26},{14,34},{14,41},{14,44},{14,64},{15,27},{15,35},{15,42},{15,45},{15,65},{16,28},{16,36},{16,43},{16,46},{16,66},{17,25},{17,29},{17,44},{17,47},{17,67},{18,26},{18,30},{18,45},{18,48},{18,68},{19,27},{19,31},{19,37},{19,46},{19,69},{20,28},{20,32},{20,38},{20,47},{20,70},{21,29},{21,33},{21,39},{21,48},{21,71},{22,30},{22,34},{22,37},{22,40},{22,72},{23,31},{23,35},{23,38},{23,41},{23,61},{24,32},{24,36},{24,39},{24,42},{24,62},{25,31},{25,62},{25,64},{26,32},{26,63},{26,65},{27,33},{27,64},{27,66},{28,34},{28,65},{28,67},{29,35},{29,66},{29,68},{30,36},{30,67},{30,69},{31,68},{31,70},{32,69},{32,71},{33,70},{33,72},{34,61},{34,71},{35,62},{35,72},{36,61},{36,63},{37,50},{37,60},{38,49},{38,51},{39,50},{39,52},{40,51},{40,53},{41,52},{41,54},{42,53},{42,55},{43,54},{43,56},{44,55},{44,57},{45,56},{45,58},{46,57},{46,59},{47,58},{47,60},{48,49},{48,59},{49,55},{50,56},{51,57},{52,58},{53,59},{54,60}};
degreediameter44 = {{1,32},{1,50},{1,66},{1,92},{2,31},{2,49},{2,65},{2,91},{3,34},{3,52},{3,68},{3,94},{4,33},{4,51},{4,67},{4,93},{5,36},{5,54},{5,70},{5,96},{6,35},{6,53},{6,69},{6,95},{7,38},{7,56},{7,58},{7,98},{8,37},{8,55},{8,57},{8,97},{9,40},{9,44},{9,60},{9,86},{10,39},{10,43},{10,59},{10,85},{11,42},{11,46},{11,62},{11,88},{12,41},{12,45},{12,61},{12,87},{13,30},{13,48},{13,64},{13,90},{14,29},{14,47},{14,63},{14,89},{15,36},{15,64},{15,74},{15,95},{16,35},{16,63},{16,73},{16,96},{17,38},{17,66},{17,76},{17,97},{18,37},{18,65},{18,75},{18,98},{19,40},{19,68},{19,78},{19,85},{20,39},{20,67},{20,77},{20,86},{21,42},{21,70},{21,80},{21,87},{22,41},{22,69},{22,79},{22,88},{23,30},{23,58},{23,82},{23,89},{24,29},{24,57},{24,81},{24,90},{25,32},{25,60},{25,84},{25,91},{26,31},{26,59},{26,83},{26,92},{27,34},{27,62},{27,72},{27,93},{28,33},{28,61},{28,71},{28,94},{29,50},{29,79},{30,49},{30,80},{31,52},{31,81},{32,51},{32,82},{33,54},{33,83},{34,53},{34,84},{35,56},{35,71},{36,55},{36,72},{37,44},{37,73},{38,43},{38,74},{39,46},{39,75},{40,45},{40,76},{41,48},{41,77},{42,47},{42,78},{43,62},{43,90},{44,61},{44,89},{45,64},{45,92},{46,63},{46,91},{47,66},{47,94},{48,65},{48,93},{49,68},{49,96},{50,67},{50,95},{51,70},{51,98},{52,69},{52,97},{53,58},{53,86},{54,57},{54,85},{55,60},{55,88},{56,59},{56,87},{57,84},{58,83},{59,72},{60,71},{61,74},{62,73},{63,76},{64,75},{65,78},{66,77},{67,80},{68,79},{69,82},{70,81},{71,90},{72,89},{73,92},{74,91},{75,94},{76,93},{77,96},{78,95},{79,98},{80,97},{81,86},{82,85},{83,88},{84,87}};
degreediameter35 = {{1,26},{1,42},{1,43},{2,30},{2,37},{2,44},{3,32},{3,34},{3,45},{4,24},{4,29},{4,46},{5,33},{5,40},{5,47},{6,35},{6,37},{6,48},{7,27},{7,32},{7,49},{8,22},{8,36},{8,50},{9,38},{9,40},{9,51},{10,30},{10,35},{10,52},{11,25},{11,39},{11,53},{12,22},{12,41},{12,54},{13,33},{13,38},{13,55},{14,28},{14,42},{14,56},{15,23},{15,25},{15,57},{16,36},{16,41},{16,58},{17,24},{17,31},{17,59},{18,26},{18,28},{18,60},{19,23},{19,39},{19,61},{20,27},{20,34},{20,62},{21,29},{21,31},{21,63},{22,43},{23,44},{24,45},{25,46},{26,47},{27,48},{28,49},{29,50},{30,51},{31,52},{32,53},{33,54},{34,55},{35,56},{36,57},{37,58},{38,59},{39,60},{40,61},{41,62},{42,63},{43,64},{44,64},{45,64},{46,65},{47,65},{48,65},{49,66},{50,66},{51,66},{52,67},{53,67},{54,67},{55,68},{56,68},{57,68},{58,69},{59,69},{60,69},{61,70},{62,70},{63,70}};
Here's a picture of degree diameter graphs with decent embeddings. Of these 6, I found all but the Petersen embedding four of these I found after posting this question.
Can anyone find nice embeddings for other graphs within this problem space?