Graphs with integer index
The following table gives the numbers of simple connected graphs of order at most 10 whose index (largest adjacency eigenvalue) is an integer. There is just one connected graph with index 1, namely K2, and exactly 17 connected graphs of order at most 10 with index 2. The latter are the well-known Smith graphs and are therefore not listed here. Together with the graphs counted in the table, these give a total of 1,328 connected graphs of order at most 10 with integer index.
| Order / index | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|---|---|
| 4 | 1 | – | – | – | – | – | – |
| 5 | 1 | 1 | – | – | – | – | – |
| 6 | 2 | 1 | 1 | – | – | – | – |
| 7 | 5 | 3 | 1 | 1 | – | – | – |
| 8 | 18 | 20 | 8 | 1 | 1 | – | – |
| 9 | 36 | 103 | 18 | 11 | 1 | 1 | – |
| 10 | 135 | 582 | 256 | 91 | 8 | 2 | 1 |
Minimal self-centred graphs
A graph is self-centred if all its vertices have the same eccentricity. The following table gives the numbers of simple minimal self-centred graphs of order at most 10. The graph K1 is self-centred and has radius 0. There are also nine minimal self-centred graphs of order at most 10 and radius 1, namely K2, K3, …, K10. Together with the graphs counted in the table, these give a total of 843 minimal self-centred graphs of order at most 10.
| Order / radius | 2 | 3 | 4 | 5 |
|---|---|---|---|---|
| 4 | 1 | – | – | – |
| 5 | 2 | – | – | – |
| 6 | 4 | 1 | – | – |
| 7 | 9 | 2 | – | – |
| 8 | 29 | 8 | 1 | – |
| 9 | 102 | 27 | 2 | – |
| 10 | 518 | 118 | 8 | 1 |
If you use these data on minimal self-centred graphs in your research, please cite the following paper: Z. Stanić, Some notes on minimal self-centered graphs, AKCE Int. J. Graphs Combin., 7 (2010), 97–102.