Connected cubic symmetric graphs are sometimes called Foster graphs and denoted 
, where 
is the vertex count and 
is a letter A, B, C, ... indicating the particular such cubic symmetric graph in the standard Foster
census enumeration an its extensions (Foster 1932, Bouwer et al. 1988, Conder
and Dobcsányi 2002, Royle). Many Foster graphs can be constructed as honeycomb
toroidal graphs.
"The" Foster graph is the cubic symmetric graph 
on 90 vertices illustrated above that has 135 edges and is also distance-regular
with intersection array 
. It has girth 10, radius 8,
and diameter 8. The Foster graph is also Hamiltonian.
It has a unique order-15 LCF notations (![[17,-9,37,-37,9,-17]^(15)](https://txtdot.deep-swarm.xyz/proxy/img?url=https%3A%2F%2Fmathworld.wolfram.com%2Fimages%2Fequations%2FFosterGraph%2FInline6.svg){kind=small}
), 4 order-5 notations, and 2 order-2
notations (illustrated above), as well as 397 order-1 notations.
The Foster graph is not uniquely determined by its graph spectrum (van Dam and Haemers 2003). It has graph spectrum
so it is not quite an integral graph.
It has chromatic number 2.
The halved Foster graph is distance-regular with intersection array 
.
See also
Cubic Graph, Cubic Symmetric Graph, Distance-Regular Graph, Foster Cage, Honeycomb Toroidal Graph, Symmetric Graph
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References
Bouwer, I. Z.; Chernoff, W. W.; Monson, B.; and Star, Z. The Foster Census. Charles Babbage Research Centre, 1988.Brouwer, A. E.; Cohen, A. M.; and Neumaier, A. Distance Regular Graphs. New York: Springer-Verlag, p. 398, 1989.Conder, M. and Dobcsányi, P. "Trivalent Symmetric Graphs Up to 768 Vertices." J. Combin. Math. Combin. Comput. 40, 41-63, 2002.Conder, M. and Potočnik, P. "Census of Cubic Edge-Transitive Graphs." https://fostercensus.graphsym.net.DistanceRegular.org. "Foster Graph." https://www.math.mun.ca/distanceregular/graphs//foster.html.DistanceRegular.org. "Halved Foster Graph." https://www.math.mun.ca/distanceregular/graphs//halved-foster.html.Foster, R. M. "Geometrical Circuits of Electrical Networks." Trans. Amer. Inst. Elec. Engin. 51, 309-317, 1932.Royle, G. "F090A." http://www.csse.uwa.edu.au/~gordon/foster/F090A.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.
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Cite this as:
Weisstein, Eric W. "Foster Graph." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/FosterGraph.html