Biggs–Smith graph
{{Short description|Cubic distance-regular graph with 102 nodes and 153 edges}}
{{infobox graph
| name = Biggs–Smith graph
| image = 250px
| image_caption = The Biggs–Smith graph
| namesake =
| vertices = 102
| edges = 153
| automorphisms= 2448 (PSL(2,17))
| girth = 9
| radius = 7
| diameter = 7
| 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.{{MathWorld|urlname=Biggs-SmithGraph|title=Biggs–Smith Graph}}
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.Brouwer, A. E.; Cohen, A. M.; and Neumaier, A. Distance-Regular Graphs. New York: Springer-Verlag, 1989. 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{{citation|url=http://atlas.gregas.eu/graphs/38|title=G-17 Biggs-Smith graph|work=Encyclopedia of graphs|access-date=2024-02-22}} 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.Conder, M. and Dobcsányi, P. "Trivalent Symmetric Graphs Up to 768 Vertices." J. Combin. Math. Combin. Comput. 40, 41–63, 2002.
The Biggs–Smith graph is also uniquely determined by its graph spectrum, the set of graph eigenvalues of its adjacency matrix.E. R. van Dam and W. H. Haemers, Spectral Characterizations of Some Distance-Regular Graphs. J. Algebraic Combin. 15, pages 189–202, 2003
The characteristic polynomial of the Biggs–Smith graph is :
.
Gallery
Image:Biggs-Smith graph 3COL.svg|The chromatic number of the Biggs–Smith graph is 3.
Image:Biggs-Smith graph 3color edge.svg|The chromatic index of the Biggs–Smith graph is 3.
Image:BiggsSmith.svg|Alternative drawing of the Biggs–Smith graph
Image:Biggs-Smith Graph.png|Another drawing of the Biggs–Smith graph
Image:Biggs-Smith graph - circle-based drawing.jpg|Decomposition of the Biggs–Smith graph into 6 sets of size 17
Image:Biggs_Smith_Graph_H_Shape.png|Another rendering of the Biggs–Smith graph, once again showing that it is an order-17 graph expansion of the H graph
References
{{reflist}}
- On trivalent graphs, NL Biggs, DH Smith - Bulletin of the London Mathematical Society, 3 (1971) 155–158.
{{DEFAULTSORT:Biggs-Smith graph}}