# Biggs–Smith graph

__: Cubic distance-regular graph with 102 nodes and 153 edges__

**Short description**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 |

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

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 : [math]\displaystyle{ (x-3) (x-2)^{18} x^{17} (x^2-x-4)^9 (x^3+3 x^2-3)^{16} }[/math].

## Gallery

The chromatic number of the Biggs–Smith graph is 3.

## References

- ↑ Weisstein, Eric W.. "Biggs–Smith Graph". http://mathworld.wolfram.com/Biggs-SmithGraph.html.
- ↑ 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.

Original source: https://en.wikipedia.org/wiki/Biggs–Smith graph.
Read more |