# Biggs–Smith graph

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Biggs–Smith graph
The Biggs–Smith graph
Vertices 102
Edges 153
Diameter 7
Girth 9
Automorphisms 2448 (PSL(2,17))
Chromatic number 3
Chromatic index 3
Properties Symmetric
Distance-regular
Cubic
Hamiltonian

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 the its graph spectrum, the set of graph eigenvalues of its adjacency matrix.[5]

The characteristic polynomial of the Biggs–Smith graph is : ${\displaystyle (x-3)(x-2)^{18}x^{17}(x^{2}-x-4)^{9}(x^{3}+3x^{2}-3)^{16}}$.

## References

1. ^
2. ^ Brouwer, A. E.; Cohen, A. M.; and Neumaier, A. Distance-Regular Graphs. New York: Springer-Verlag, 1989.
3. ^ Royle, G. F102A data
4. ^ Conder, M. and Dobcsányi, P. "Trivalent Symmetric Graphs Up to 768 Vertices." J. Combin. Math. Combin. Comput. 40, 41–63, 2002.
5. ^ 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.