Biggs–Smith graph

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

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 : (x-3) (x-2)^{18} x^{17} (x^2-x-4)^9 (x^3%2B3 x^2-3)^{16}.

Gallery

References

  1. ^ Weisstein, Eric W., "Biggs–Smith Graph" from MathWorld.
  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