The next term a(24) corresponding to the 6-regular graphs on 15 nodes is conjectured to be 1. It seems that there exists only one graph with diameter A294733(24)=4. Its adjacency matrix is
1 2 3 4 5 6 7 8 9 0 1 2 3 4 5
1 . 1 1 1 1 1 1 . . . . . . . .
2 1 . 1 1 1 1 1 . . . . . . . .
3 1 1 . 1 1 1 1 . . . . . . . .
4 1 1 1 . 1 1 1 . . . . . . . .
5 1 1 1 1 . 1 1 . . . . . . . .
6 1 1 1 1 1 . . 1 . . . . . . .
7 1 1 1 1 1 . . 1 . . . . . . .
8 . . . . . 1 1 . 1 1 1 1 . . .
9 . . . . . . . 1 . 1 1 . 1 1 1
10 . . . . . . . 1 1 . . 1 1 1 1
11 . . . . . . . 1 1 . . 1 1 1 1
12 . . . . . . . 1 . 1 1 . 1 1 1
13 . . . . . . . . 1 1 1 1 . 1 1
14 . . . . . . . . 1 1 1 1 1 . 1
15 . . . . . . . . 1 1 1 1 1 1 .
The distance of 4 is achieved between nodes 1 and 13. None of the remaining 1470293674 graphs seems to have a diameter > 3.
The conjecture is confirmed using Markus Meringer's GenReg program. Aside from the 1 shown 6-regular graph on 15 nodes with diameter 4 there are 870618932 graphs with diameter 2 and 599674742 graphs with diameter 3. - Hugo Pfoertner, Dec 19 2017
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