Foster graphs are connected cubic symmetric graphs and are denoted vertex count and
cubic symmetric graph
in the standard Foster census enumeration and 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 vertices illustrated above that has 135 edges and is also distance-regular
with intersection array girth
10, graph radius 8, and graph
diameter 8. The Foster graph is also Hamiltonian .
It has a unique order-15 LCF notation (LCF
notations , and 2 order-2 LCF notations (illustrated
above), as well as 397 order-1 LCF 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 ,
Half-Arc-Transitive
Graph ,
Honeycomb Toroidal Graph ,
Line Graph ,
Symmetric Graph
Explore with Wolfram|Alpha
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. 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. House of Graphs. "Foster
Graph." https://houseofgraphs.org/graphs/37261 . Royle,
G. "F090A." https://web.archive.org/web/20081120121408/http://people.csse.uwa.edu.au/gordon/remote/foster/F090A.html . Royle,
G. "Cubic Symmetric Graphs (the Foster Census): Distance-Regular Graphs."
https://web.archive.org/web/20081004205049/http://people.csse.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. Referenced
on Wolfram|Alpha Foster Graph
Cite this as:
Weisstein, Eric W. "Foster Graph." From
MathWorld https://mathworld.wolfram.com/FosterGraph.html
Subject classifications More... Less...
, 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 and its extensions (Foster 1932, Bouwer
et al. 1988, Conder and Dobcsányi 2002, Royle). Many 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, graph radius 8, and graph
diameter 8. The Foster graph is also Hamiltonian.
It has a unique order-15 LCF notation (![[17,-9,37,-37,9,-17]^(15)](/images/equations/FosterGraph/Inline6.svg)
), 4 order-5 LCF
notations, and 2 order-2 LCF notations (illustrated
above), as well as 397 order-1 LCF notations.
.
