Some of the finite structures considered in graph theory have names, sometimes inspired by the graph's topology, and sometimes after their discoverer. A famous example is the Petersen graph, a concrete graph on 10 vertices that appears as a minimal example or counterexample in many different contexts.
Individual graphs
Balaban 10-cage
Balaban 11-cage
Bidiakis cube
Brinkmann graph
Bull graph
Butterfly graph
Chvátal graph
Diamond graph
Dürer graph
Ellingham–Horton 54-graph
Ellingham–Horton 78-graph
Errera graph
Franklin graph
Frucht graph
Goldner–Harary graph
Golomb graph
Harries graph
Harries–Wong graph
Herschel graph
Hoffman graph
Holt graph
Horton graph
Kittell graph
Markström graph
McGee graph
Meredith graph
Moser spindle
Sousselier graph
Poussin graph
Robertson graph
Sylvester graph
Tutte's fragment
Tutte graph
Young–Fibonacci graph
Wagner graph
Wells graph
Wiener–Araya graph
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Highly symmetric graphs
Strongly regular graphs
The strongly regular graph on v vertices and rank k is usually denoted srg(v,k,λ,μ).
Clebsch graph
Cameron graph
Hall–Janko graph
Hoffman–Singleton graph
Higman–Sims graph
Paley graph of order 13
Schläfli graph
Brouwer–Haemers graph
Local McLaughlin graph
Perkel graph
Gewirtz graph
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Symmetric graphs
A symmetric graph is one in which there is a symmetry (graph automorphism) taking any ordered pair of adjacent vertices to any other ordered pair; the Foster census lists all small symmetric 3-regular graphs. Every strongly regular graph is symmetric, but not vice versa.
Möbius–Kantor graph
Nauru graph
Coxeter graph
Tutte–Coxeter graph
Dyck graph
Klein graph
Biggs–Smith graph
The Rado graph
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Semi-symmetric graphs
Folkman graph
Ljubljana graph
Tutte 12-cage
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Graph families
Complete graphs
The complete graph on vertices is often called the -clique and usually denoted , from German komplett.[1]
Complete bipartite graphs
The complete bipartite graph is usually denoted . For see the section on star graphs. The graph equals the 4-cycle (the square) introduced below.
, the utility graph
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Cycles
The cycle graph on vertices is called the n-cycle and usually denoted . It is also called a cyclic graph, a polygon or the n-gon. Special cases are the triangle , the square , and then several with Greek naming pentagon , hexagon , etc.
Friendship graphs
The friendship graph Fn can be constructed by joining n copies of the cycle graph C3 with a common vertex.[2]
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Fullerene graphs
In graph theory, the term fullerene refers to any 3-regular, planar graph with all faces of size 5 or 6 (including the external face). It follows from Euler's polyhedron formula, V – E + F = 2 (where V, E, F indicate the number of vertices, edges, and faces), that there are exactly 12 pentagons in a fullerene and h = V/2 – 10 hexagons. Therefore V = 20 + 2h; E = 30 + 3h. Fullerene graphs are the Schlegel representations of the corresponding fullerene compounds.
20-fullerene (dodecahedral graph)
24-fullerene (Hexagonal truncated trapezohedron graph)
26-fullerene graph
60-fullerene (truncated icosahedral graph)
70-fullerene
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An algorithm to generate all the non-isomorphic fullerens with a given number of hexagonal faces has been developed by G. Brinkmann and A. Dress.[3] G. Brinkmann also provided a freely available implementation, called fullgen.
Platonic solids
The complete graph on four vertices forms the skeleton of the tetrahedron, and more generally the complete graphs form skeletons of simplices. The hypercube graphs are also skeletons of higher-dimensional regular polytopes.
Cube
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Truncated solids
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Snarks
A snark is a bridgeless cubic graph that requires four colors in any proper edge coloring. The smallest snark is the Petersen graph, already listed above.
Blanuša snark (first)
Blanuša snark (second)
Double-star snark
Flower snark
Loupekine snark (first)
Loupekine snark (second)
Tietze graph
Watkins snark
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Star
A star Sk is the complete bipartite graph K1,k. The star S3 is called the claw graph.
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Wheel graphs
The wheel graph Wn is a graph on n vertices constructed by connecting a single vertex to every vertex in an (n − 1)-cycle.
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References
- ^ David Gries and Fred B. Schneider, A Logical Approach to Discrete Math, Springer, 1993, p 436.
- ^ Gallian, J. A. "Dynamic Survey DS6: Graph Labeling." Electronic Journal of Combinatorics, DS6, 1-58, January 3, 2007. [1] Archived 2012-01-31 at the Wayback Machine.
- ^ Brinkmann, Gunnar; Dress, Andreas W.M (1997). "A Constructive Enumeration of Fullerenes". Journal of Algorithms. 23 (2): 345–358. doi:10.1006/jagm.1996.0806. MR 1441972.