Gallery of named graphs

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.

See also: Glossary of graph theory terms

Not to be confused with Named graph.

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
• 
  Grötzsch 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

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
• 
  Petersen graph
• 
  Hall–Janko graph
• 
  Hoffman–Singleton graph
• 
  Higman–Sims graph
• 
  Paley graph of order 13
• 
  Shrikhande graph
• 
  Schläfli graph
• 
  Brouwer–Haemers graph
• 
  Local McLaughlin graph
• 
  Perkel graph
• 
  Gewirtz graph

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.

• 
  Heawood graph
• 
  Möbius–Kantor graph
• 
  Pappus graph
• 
  Desargues graph
• 
  Nauru graph
• 
  Coxeter graph
• 
  Tutte–Coxeter graph
• 
  Dyck graph
• 
  Klein graph
• 
  Foster graph
• 
  Biggs–Smith graph
• 
  The Rado graph

Semi-symmetric graphs

• 
  Folkman graph
• 
  Gray graph
• 
  Ljubljana graph
• 
  Tutte 12-cage

Graph families

Complete graphs

The complete graph on ${\displaystyle n}$ vertices is often called the ${\displaystyle n}$-clique and usually denoted ${\displaystyle K_{n}}$, from German komplett.[1]

• 
  ${\displaystyle K_{1}}$
• 
  ${\displaystyle K_{2}}$
• 
  ${\displaystyle K_{3}}$
• 
  ${\displaystyle K_{4}}$
• 
  ${\displaystyle K_{5}}$
• 
  ${\displaystyle K_{6}}$
• 
  ${\displaystyle K_{7}}$
• 
  ${\displaystyle K_{8}}$

Complete bipartite graphs

The complete bipartite graph is usually denoted ${\displaystyle K_{n,m}}$. For ${\displaystyle n=1}$ see the section on star graphs. The graph ${\displaystyle K_{2,2}}$ equals the 4-cycle ${\displaystyle C_{4}}$ (the square) introduced below.

• 
  ${\displaystyle K_{2,3}}$
• 
  ${\displaystyle K_{3,3}}$, the utility graph
• 
  ${\displaystyle K_{2,4}}$
• 
  ${\displaystyle K_{3,4}}$

Cycles

The cycle graph on ${\displaystyle n}$ vertices is called the n-cycle and usually denoted ${\displaystyle C_{n}}$. It is also called a cyclic graph, a polygon or the n-gon. Special cases are the triangle ${\displaystyle C_{3}}$, the square ${\displaystyle C_{4}}$, and then several with Greek naming pentagon ${\displaystyle C_{5}}$, hexagon ${\displaystyle C_{6}}$, etc.

• 
  ${\displaystyle C_{3}}$
• 
  ${\displaystyle C_{4}}$
• 
  ${\displaystyle C_{5}}$
• 
  ${\displaystyle C_{6}}$

Friendship graphs

The friendship graph F_n can be constructed by joining n copies of the cycle graph C₃ with a common vertex.[2]

The friendship graphs F₂, F₃ and F₄.

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

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
  ${\displaystyle n=8}$, ${\displaystyle m=12}$
• 
  Octahedron
  ${\displaystyle n=6}$, ${\displaystyle m=12}$
• 
  Dodecahedron
  ${\displaystyle n=20}$, ${\displaystyle m=30}$
• 
  Icosahedron
  ${\displaystyle n=12}$, ${\displaystyle m=30}$

Truncated solids

• 
  Truncated tetrahedron
• 
  Truncated cube
• 
  Truncated octahedron
• 
  Truncated dodecahedron
• 
  Truncated icosahedron

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)
• 
  Szekeres snark
• 
  Tietze graph
• 
  Watkins snark

Star

A star S_k is the complete bipartite graph K_1,k. The star S₃ is called the claw graph.

The star graphs S₃, S₄, S₅ and S₆.

Wheel graphs

The wheel graph W_n is a graph on n vertices constructed by connecting a single vertex to every vertex in an (n − 1)-cycle.

Wheels ${\displaystyle W_{4}}$ – ${\displaystyle W_{9}}$.

References

1. David Gries and Fred B. Schneider, A Logical Approach to Discrete Math, Springer, 1993, p 436.
2. Gallian, J. A. "Dynamic Survey DS6: Graph Labeling." Electronic Journal of Combinatorics, DS6, 1-58, January 3, 2007. Archived 2012-01-31 at the Wayback Machine.
3. 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.