# Biggs–Smith graph

In the mathematical field of graph theory, the **Biggs–Smith graph** is a 3-regular graph with 102 vertices and 153 edges.[1]

Biggs–Smith graph | |
---|---|

The Biggs–Smith graph | |

Vertices | 102 |

Edges | 153 |

Radius | 7 |

Diameter | 7 |

Girth | 9 |

Automorphisms | 2448 (PSL(2,17)) |

Chromatic number | 3 |

Chromatic index | 3 |

Properties | Symmetric Distance-regular Cubic Hamiltonian |

Table of graphs and parameters |

It has chromatic number 3, chromatic index 3, radius 7, diameter 7 and girth 9. It is also a 3-vertex-connected graph and a 3-edge-connected graph.

All the cubic distance-regular graphs are known.[2] The Biggs–Smith graph is one of the 13 such graphs.

## Algebraic properties

The automorphism group of the Biggs–Smith graph is a group of order 2448[3] isomorphic to the projective special linear group PSL(2,17). It acts transitively on the vertices, on the edges and on the arcs of the graph. Therefore, the Biggs–Smith 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 Biggs–Smith graph, referenced as F102A, is the only cubic symmetric graph on 102 vertices.[4]

The Biggs–Smith graph is also uniquely determined by its graph spectrum, the set of graph eigenvalues of its adjacency matrix.[5]

The characteristic polynomial of the Biggs–Smith graph is : .

## Gallery

## References

- Weisstein, Eric W. "Biggs–Smith Graph".
*MathWorld*. - Brouwer, A. E.; Cohen, A. M.; and Neumaier, A. Distance-Regular Graphs. New York: Springer-Verlag, 1989.
- Royle, G. F102A data
- Conder, M. and Dobcsányi, P. "Trivalent Symmetric Graphs Up to 768 Vertices." J. Combin. Math. Combin. Comput. 40, 41–63, 2002.
- E. R. van Dam and W. H. Haemers, Spectral Characterizations of Some Distance-Regular Graphs. J. Algebraic Combin. 15, pages 189–202, 2003

- On trivalent graphs, NL Biggs, DH Smith - Bulletin of the London Mathematical Society, 3 (1971) 155-158.