The complete graph is symmetrical.
In addition, if the number of vertices is > 3, the simple cycle through all vertices is symmetrical.
Graphs determined by vertices and edges of Platonic solids are symmetrical.
The square K X K grid with right vertices connected to corresponding left vertices and bottom vertices connected to corresponding top vertices is symmetrical.
The smallest nontrivial and nonPlatonic symmetric graph is the hexagon with connected opposite vertices.
An example of symmetrical graph with 13 vertices:
0 connected to 1, 2, 3, 4
1 connected to 0, 5, 6, 7
2 connected to 0, 5, 8, 9
3 connected to 0, 6, 10, 11
4 connected to 0, 8, 10, 12
5 connected to 1, 2, 10, 11
6 connected to 1, 3, 8, 12
7 connected to 1, 8, 9, 11
8 connected to 2, 4, 6, 7
9 connected to 2, 7, 10, 12
10 connected to 3, 4, 5, 9
11 connected to 3, 5, 7, 12
12 connected to 4, 6, 9, 11
