TOPICS

# Biggs-Smith Graph

The Biggs-Smith graph is cubic symmetric graph on 102 vertices and 153 edges that is also distance-regular with intersection array and distance-transitive.

It is known to be uniquely determined by its graph spectrum (van Dam and Haemers 2003). Its automorphism group is of order 2448 (Royle).

It is implemented in the Wolfram Language as GraphData["BiggsSmithGraph"].

The Biggs-Smith graph is an order-17 graph expansion of the H graph with step offsets 3, 5, 6, and 7 (where these are a different set of steps from those reported by Biggs 1993, p. 147). It is therefore one of only two cubic symmetric H graphs (the other being ).

The Biggs-Smith graph is a unit-distance graph, as are all cubic symmetric H-, I-, and Y-graphs (E. Gerbracht, pers. comm., Jan. 2010).

The Biggs-Smith graph has distinct (directed) Hamiltonian cycles which correspond to 890 distinct LCF notations, all of which are of order 1 (E. Weisstein, May 30, 2008). One such LCF notation (of length 102) is given by [16, 24, , 17, 34, 48, , 41, , 47, , 34, , 21, 14, 48, , , , 28, , 21, 29, , 46, , 28, , , , , 21, 8, 27, , 20, , 39, , , , 38, , 25, 15, , 18, , , 36, 8, , , , , , , 14, , , , 38, 24, , , 25, 38, 31, , 24, , , 28, 11, 21, 35, , 43, 36, , 14, 50, 43, 36, , , , 45, 8, 19, , 38, 20, , , , , 44, , , , 37].

The plots above show the adjacency, incidence, and distance matrices of the graph.

The bipartite double graph and double cover of the Biggs-Smith graph is the cubic symmetric graph .

Cubic Symmetric Graph, Distance-Regular Graph, Graph Expansion

## Explore with Wolfram|Alpha

More things to try:

## References

Biggs, N. L. Algebraic Graph Theory, 2nd ed. Cambridge, England: Cambridge University Press, 1993.DistanceRegular.org. "Biggs-Smith Graph." http://www.distanceregular.org/graphs/biggssmith.html.Royle, G. "F102A." http://www.csse.uwa.edu.au/~gordon/foster/F102A.html.Royle, G. "Cubic Symmetric Graphs (The Foster Census): Distance-Regular Graphs." http://school.maths.uwa.edu.au/~gordon/remote/foster/#drgs.van Dam, E. R. and Haemers, W. H. "Spectral Characterizations of Some Distance-Regular Graphs." J. Algebraic Combin. 15, 189-202, 2003.Van Maldeghem, H. and Ver Gucht, V. "Some Properties of the Biggs-Smith Geometry." Bull. Belg. Math. Soc. Simon Stevin 12, 919-924, 2006.

## Cite this as:

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