Graphs with integer index
The following table contains numbers of simple connected graphs with at most 10 vertices and integer index. There is only one connected graph with index 1 (it is K2) and exactly 17 connected graphs with index 2 and and no more than 10 vertices (these graphs are well known as Smith graphs and we omit their presentation here). Combining these values with the following table, we obtain the 1328 graphs with integer index.
Minimal self-centered graphs
A graph is self-centered if all vertices have equal eccentricity. The following table contains numbers of simple minimal self-centered graphs with at most 10 vertices. Graph K1 is self-centered (with radius 0). Also, there are 9 minimal self-centered graphs with radius 1 and no more than 10 vertices (K2 - K10). Combining these values with the following table, we obtain the 843 minimal self-centered graphs.
If you use the above data on minimal self-centered graph in your research, please cite this paper: Z. Stanić, Some notes on minimal self-centered graphs, AKCE Int. J. Graphs Combin., 7 (2010) 97-102.