A is a ring of 8 nodes evenly spaced on a circle. Each node is connected to the next node in the ring and the node (n+3) % 8. Note that none of the edges pass through the center point of the circle.
B is a ring of 8 nodes evenly spaced on a circle. Each node is connected to the next node in the ring and the node (n+4) % 8. Note that each edge not on the outer ring passes through the center of the circle.
C is a ring of 10 evenly spaced nodes where the calculation of non-ring edges is (n+5) % 10.
For any ring with an even number of nodes it appears that if you connect a node with a next node given by (n+(number of nodes / 2)) % number of nodes you end up with a graph where the non-ring edges pass through the center of the circle.
What is this property called?
Note: I'm a total novice so apologies if this is easy to search for given elementary terminology.
These are special cases of circulant graphs (Wikipedia; MathWorld). Specifically, they are circulant graphs with offset list $\{1, \lfloor n/2 \rfloor\}$ (depending on your choice of notation and nomenclature).
Moreover, B and C are examples of wheel graphs (Wikipedia). I don't know of a special name for A-like graphs.