Given any tree having vertices of vertex degrees of 1 and 3 only, form an -expansion by taking disjoint copies of and joining corresponding leaves by an -cycle where, however, the th leaf on the th copy need not be connected to the th leaf on the st copy, but will in general be connected to the th copy. The set of values are known as the steps.

The resulting graphs are always cubic, and there exist exactly 13 graph expansions that are symmetric
as well, as summarized in the following table (Biggs 1993, p. 147). E. Gerbracht
(pers. comm., Jan. 29, 2010) has shown that all the graphs in this table are
unit-distance.

Biggs, N. L. Algebraic Graph Theory, 2nd ed. Cambridge, England: Cambridge University Press, p. 147,
1993.Horton, J. D. and Bouwer, I. Z. "Symmetric Y-Graphs
and H-Graphs." J. Combin. Th. Ser. B53, 114-129, 1991.