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 (if
not all) 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 (), 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
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.DistanceRegular.org.
"Foster Graph." http://www.distanceregular.org/graphs/foster.html.DistanceRegular.org.
"Halved Foster Graph." http://www.distanceregular.org/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.
,
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 (if
not all) Foster graphs can be constructed as honeycomb
toroidal graphs.
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)](/images/equations/FosterGraph/Inline6.svg)
), 4 order-5 notations, and 2 order-2
notations (illustrated above), as well as 397 order-1 notations.
.
