# 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 after | Ronald Martin Foster |

Vertices | 90 |

Edges | 135 |

Radius | 8 |

Diameter | 8 |

Girth | 10 |

Automorphisms | 4320 |

Chromatic number | 2 |

Chromatic index | 3 |

Queue number | 2 |

Properties | Cubic 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 .

## Gallery

## References

- Weisstein, Eric W. "Foster Graph".
*MathWorld*. - Wolz, Jessica;
*Engineering Linear Layouts with SAT.*Master Thesis, University of Tübingen, 2018 - Brouwer, A. E.; Cohen, A. M.; and Neumaier, A. Distance-Regular Graphs. New York: Springer-Verlag, 1989.
- Cubic distance-regular graphs, A. Brouwer.
- Hiraki, Akira; Nomura, Kazumasa; Suzuki, Hiroshi (2000), "Distance-regular graphs of valency 6 and ",
*Journal of Algebraic Combinatorics*,**11**(2): 101–134, doi:10.1023/A:1008776031839, MR 1761910 - Royle, G. F090A data
- 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.