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.
Highly symmetric graphsEdit
Strongly regular graphsEdit
The strongly regular graph on v vertices and rank k is usually denoted srg(v,k,λ,μ).
Paley graph of order 13
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.
The Rado graph
Complete bipartite graphsEdit
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
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.
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)
60-fullerene (truncated icosahedral graph)
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. G. Brinkmann also provided a freely available implementation, called fullgen.
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.
The wheel graph Wn is a graph on n vertices constructed by connecting a single vertex to every vertex in an (n − 1)-cycle.
- 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.  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.