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

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

Gear

G₄

A gear graph, denoted G_n, is a graph obtained by inserting an extra vertex between each pair of adjacent vertices on the perimeter of a wheel graph W_n. Thus, G_n has 2n+1 vertices and 3n edges.^[4] Gear graphs are examples of squaregraphs, and play a key role in the forbidden graph characterization of squaregraphs.^[5] Gear graphs are also known as cogwheels and bipartite wheels.

Helm

A helm graph, denoted H_n, is a graph obtained by attaching a single edge and node to each node of the outer circuit of a wheel graph W_n.^[6][7]

Lobster

A lobster graph is a tree in which all the vertices are within distance 2 of a central path.^[8][9] Compare caterpillar.

Web

The web graph W_4,2 is a cube.

The web graph W_n,r is a graph consisting of r concentric copies of the cycle graph C_n, with corresponding vertices connected by "spokes". Thus W_n,1 is the same graph as C_n, and W_n,2 is a prism.

A web graph has also been defined as a prism graph Y_{n+1, 3}, with the edges of the outer cycle removed.^[7][10]