# Foster graph

In the mathematical field of graph theory, the Foster graph is a bipartite 3-regular graph with 90 vertices and 135 edges.[1]

Foster graph
The Foster graph
Named afterRonald Martin Foster
Vertices90
Edges135
Diameter8
Girth10
Automorphisms4320
Chromatic number2
Chromatic index3
Queue number2
PropertiesCubic
Bipartite
Symmetric
Hamiltonian
Distance-transitive
Table of graphs and parameters

The Foster graph is Hamiltonian and has chromatic number 2, chromatic index 3, radius 8, diameter 8 and girth 10. It is also a 3-vertex-connected and 3-edge-connected graph. It has queue number 2 and the upper bound on the book thickness is 4.[2]

All the cubic distance-regular graphs are known.[3] The Foster graph is one of the 13 such graphs. It is the unique distance-transitive graph with intersection array {3,2,2,2,2,1,1,1;1,1,1,1,2,2,2,3}.[4] It can be constructed as the incidence graph of the partial linear space which is the unique triple cover with no 8-gons of the generalized quadrangle GQ(2,2). It is named after R. M. Foster, whose Foster census of cubic symmetric graphs included this graph.

The bipartite half of the Foster graph is a distance-regular graph and a locally linear graph. It is one of a finite number of such graphs with degree six.[5]

## Algebraic properties

The automorphism group of the Foster graph is a group of order 4320.[6] It acts transitively on the vertices, on the edges and on the arcs of the graph. Therefore, the Foster graph is a symmetric graph. It has automorphisms that take any vertex to any other vertex and any edge to any other edge. According to the Foster census, the Foster graph, referenced as F90A, is the only cubic symmetric graph on 90 vertices.[7]

The characteristic polynomial of the Foster graph is equal to ${\displaystyle (x-3)(x-2)^{9}(x-1)^{18}x^{10}(x+1)^{18}(x+2)^{9}(x+3)(x^{2}-6)^{12}}$.

## References

1. Weisstein, Eric W. "Foster Graph". MathWorld.
2. Wolz, Jessica; Engineering Linear Layouts with SAT. Master Thesis, University of Tübingen, 2018
3. Brouwer, A. E.; Cohen, A. M.; and Neumaier, A. Distance-Regular Graphs. New York: Springer-Verlag, 1989.
4. Cubic distance-regular graphs, A. Brouwer.
5. Hiraki, Akira; Nomura, Kazumasa; Suzuki, Hiroshi (2000), "Distance-regular graphs of valency 6 and ${\displaystyle a_{1}=1}$", Journal of Algebraic Combinatorics, 11 (2): 101–134, doi:10.1023/A:1008776031839, MR 1761910
6. Royle, G. F090A data
7. Conder, M. and Dobcsányi, P. "Trivalent Symmetric Graphs Up to 768 Vertices." J. Combin. Math. Combin. Comput. 40, 41-63, 2002.
• Biggs, N. L.; Boshier, A. G.; Shawe-Taylor, J. (1986), "Cubic distance-regular graphs", Journal of the London Mathematical Society, 33 (3): 385–394, doi:10.1112/jlms/s2-33.3.385, MR 0850954.
• Van Dam, Edwin R.; Haemers, Willem H. (2002), "Spectral characterizations of some distance-regular graphs", Journal of Algebraic Combinatorics, 15 (2): 189–202, doi:10.1023/A:1013847004932, MR 1887234.
• Van Maldeghem, Hendrik (2002), "Ten exceptional geometries from trivalent distance regular graphs", Annals of Combinatorics, 6 (2): 209–228, doi:10.1007/PL00012587, MR 1955521.