This is the Harries graph. It is a graph with the minimum number of vertices such that each vertex is connected to 3 others and every cycle has length at least 10. Such graphs are called (3,10)-cages.

This is the Balaban 10-cage, the first known (3,10)-cage. An $(r,g)$-cage is graph where every vertex has $r$ neighbors, the shortest cycle has length at least $g$, and the number of vertices is maximal given these constraints.