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# Graph Expansion

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.

 graph Foster generalized Petersen graph base graph expansion 8 cubical graph I graph (4; 1, 1) 10 Petersen graph I graph (5; 1, 2) 16 Möbius-Kantor graph I graph (8; 1, 3) 20 dodecahedral graph I graph (10; 1, 2) 20 Desargues graph I graph (10; 1, 3) 24 Nauru graph I graph (12; 1, 5) 28 Coxeter graph Y graph (7; 1, 2, 4) 48 cubic symmetric graph I graph (24; 1, 5) 56 cubic symmetric graph Y graph (14; 1, 3, 5) 102 Biggs-Smith graph H graph (17; 3, 5, 6, 7) 112 cubic symmetric graph Y graph (28; 1, 3, 9) 204 cubic symmetric graph H graph (34; 3, 5, 7, 11) 224 cubic symmetric graph Y graph (56; 1, 9, 25)

Biggs-Smith Graph, Edge Contraction, H Graph, I Graph, Y Graph

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## References

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. B 53, 114-129, 1991.

Graph Expansion

## Cite this as:

Weisstein, Eric W. "Graph Expansion." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GraphExpansion.html