This page contains some collections of graphs. See the data formats page for how to use them.

Because there are so many graphs on 11 vertices, we provide them in incremental sparse6 format. That is not yet available in showg but you can read it with the listg utility in the nauty package and convert it to other formats with the copyg utility. However, if you have nauty installed, why not just make the graphs yourself using the geng utility?

2 vertices:
all (2)
connected (1)

3 vertices:
all (4)
connected (2)

4 vertices:
all (11)
connected (6)

5 vertices:
all (34)
connected (21)

6 vertices:
all (156)
connected (112)

7 vertices:
all (1044)
connected (853)

8 vertices:
all (12346)
connected (11117)

9 vertices:
all (274668)
connected (261080)

10 vertices:
all (31MB gzipped) (12005168)
connected (30MB gzipped) (11716571)

11 vertices:
all (373MB gzipped) (1018997864)

Next we give simple graphs by their number of edges, not allowing isolated vertices but allowing disconnected graphs.

1 edge (1)

2 edges (2)

3 edges (5)

4 edges (11)

5 edges (26)

6 edges (68)

7 edges (177)

8 edges (497)

9 edges (1476)

10 edges (4613)

11 edges (15216)

12 edges (52944)

13 edges (193367)

14 edges (740226)

15 edges (2960520)

16 edges (12334829)

17 edges (53394755, gzipped)

Next we give simple connected graphs by their number of edges.

1 edge (1)

2 edges (1)

3 edges (3)

4 edges (5)

5 edges (12)

6 edges (30)

7 edges (79)

8 edges (227)

9 edges (710)

10 edges (2322)

11 edges (8071)

12 edges (29503)

13 edges (112822)

14 edges (450141)

15 edges (1867871)

16 edges (8037472)

17 edges (35787667, gzipped)

18 edges (164551477, gzipped)

The above graphs, and many varieties of them, can be efficiently generated using the program geng.

A table giving the number of graphs according to the number of edges and vertices, up to 30 vertices, can be found here.

Here we give the small simple graphs with every degree even.

2 vertices:
all (1)

3 vertices:
all (2)
connected (1)

4 vertices:
all (3)
connected (1)

5 vertices:
all (7)
connected (4)

6 vertices:
all (16)
connected (8)

7 vertices:
all (54)
connected (37)

8 vertices:
all (243)
connected (184)

9 vertices:
all (2038)
connected (1782)

10 vertices:
all (33120)
connected (31026)

11 vertices:
all (1182004)
connected (1148626)

12 vertices:
part 1;
part 2;
part 3;
part 4;
(each file about 81MB)
(87723296)

A graph is *chordal* if every cycle of length at least 4 has a chord.
Here are some files of connected chordal graphs.

4 vertices (5)

5 vertices (15)

6 vertices (58)

7 vertices (272)

8 vertices (1614)

9 vertices (11911)

10 vertices (109539)

11 vertices (1247691)

12 vertices (17566431, gzipped)

13 vertices (305310547, gzipped)

A graph is *perfect* if every odd cycle of length at least 5 has a chord,
and the same is true of the complement graph.
Here are some files of perfect graphs.

5 vertices (33)

6 vertices (148)

7 vertices (906)

8 vertices (8887)

9 vertices (136756)

10 vertices (3269264)

11 vertices (115811998, gzipped)

Here are some strongly regular graphs made by myself and/or Ted Spence and/or someone else. More information and more graphs can be found on Ted's strongly-regular page.

SRG(25,8,3,2) (1 graph)

SRG(25,12,5,6) (15 graphs)

SRG(26,10,3,4) (10 graphs)

SRG(27,10,1,5) (1 graph)

SRG(28,12,6,4) (4 graphs)

SRG(29,14,6,7) (41 graphs)

SRG(35,16,6,8) (3854 graphs)

SRG(35,18,9,9) (227 graphs)

SRG(36,14,4,6) (180 graphs)

SRG(36,15,6,6) (32548 graphs, gzipped).
These come in 227 switching classes, one for each regular two-graph
of order 36. We also provide
one representative of each class.

SRG(37,18,8,9) (6760 graphs,
maybe incomplete)

SRG(40,12,2,4) (28 graphs)

SRG(65,32,15,16) (32 graphs, maybe incomplete).
The existence was an open problem for a long time until
Oleg Gritsenko found one.
We have expanded the set to 32 graphs but there may be more.

A *Ramsey(s,t)-graph* is a graph with no clique of size s,
and no independent set of size t. On the Ramsey
graph page we present some of these graphs.

A graph is *hypohamiltonian* if it is not Hamiltonian but
each graph that can be formed from it by removing one vertex is
Hamiltonian. The smallest is the Petersen graph. The following
are all hypohamiltonian graphs with fewer than 18 vertices,
and a selection of larger hypohamiltonian graphs.

10 vertices (1 graph)

13 vertices (1 graph)

15 vertices (1 graph)

16 vertices (4 graphs)

18 vertices (13 graphs, maybe incomplete)

22 vertices (10 graphs, maybe incomplete)

26 vertices (2033 graphs, maybe incomplete)

In the case of hypohamiltonian cubic graphs we can give a complete catalogue to a larger size. Up to 26 vertices inclusive we give all of them. For 28 vertices we give those with girth at least 5, and for 30 vertices girth at least 6.

10 vertices (1 graph)

18 vertices (2 graphs)

20 vertices (1 graph)

22 vertices (3 graphs)

24 vertices (1 graph)

26 vertices (100 graphs)

28 vertices (34 graphs)

30 vertices (1 graph)

Here are give some non-isomorphic connected planar graphs. Isomorphism is according to the combinatorial structure regardless of embeddings. If you are looking for planar graphs embedded in the plane in all possible ways, your best option is to generate them using plantri.

1 vertex (1 graph)

2 vertices (1 graph)

3 vertices (2 graphs)

4 vertices (6 graphs)

5 vertices (20 graphs)

6 vertices (99 graphs)

7 vertices (646 graphs)

8 vertices (5974 graphs)

9 vertices (71885 graphs)

10 vertices (gzipped) (1052805 graphs)

11 vertices (gzipped)
Part A
Part B
(17449299 graphs)

Also see the Plane graphs page.

On the regular page we provide many counts of labelled regular graphs.

On the semiregular page we provide many counts of labelled semiregular bipartite graphs.

A self-complementary graph is one isomorphic to its complement. Such graphs can only have orders congruent to 0 or 1 modulo 4.

4 vertices (1 graph)

5 vertices (2 graphs)

8 vertices (10 graphs)

9 vertices (36 graphs)

12 vertices (720 graphs)

13 vertices (5600 graphs)

16 vertices (gzipped) (703760 graphs)

17 vertices (gzipped)
Part A
Part B
Part C (11220000 graphs)

20 vertices (incomplete, gzipped)
Part A
Part B
Part C
Part D (8571844 graphs)

The 20-vertex graphs provided are those which have a complementing permutation of order 8 or 16. There is a much larger number of graphs with complementing permutations of order 4. The total count for order 20 is 9168331776, which is too many to present here. The number of self-complementary graphs of order 21 is 293293716992.

A connected graph is highly irregular if the neighbours of each vertex have distinct degrees. Such graphs exist on all orders except 3, 5 and 7.

1 vertex (1 graph)

2 vertices (1 graph)

4 vertices (1 graph)

6 vertices (1 graph)

8 vertices (3 graphs)

9 vertices (3 graphs)

10 vertices (13 graphs)

11 vertices (21 graphs)

12 vertices (110 graphs)

13 vertices (474 graphs)

14 vertices (2545 graphs)

15 vertices (18696 graphs)

We will call an undirected simple graph *G* with no
isolated vertices *edge-k-critical* if it has chromatic number
*k* and, for every edge *e*, *G-e* has chromatic
number *k*-1. It is easy to check that *G* must be connected.

The following graphs were generated by Olivier Lalonde. Olivier's code can be found on Github. Files with more than a million graphs are gzipped.

k |
|||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

n |
4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | |

4 | 1 | ||||||||||

5 | 0 | 1 | |||||||||

6 | 1 | 0 | 1 | ||||||||

7 | 2 | 1 | 0 | 1 | |||||||

8 | 5 | 2 | 1 | 0 | 1 | ||||||

9 | 21 | 21 | 2 | 1 | 0 | 1 | |||||

10 | 150 | 162 | 22 | 2 | 1 | 0 | 1 | ||||

11 | 1221 | 4008 | 393 | 22 | 2 | 1 | 0 | 1 | |||

12 | 14581 | 147753 | 17036 | 395 | 22 | 2 | 1 | 0 | 1 | ||

13 | 207969 | 8311809 | 1479809 | 25355 | 395 | 22 | 2 | 1 | 0 | 1 | |

14 | 3567180 |

A graph with vertices 0,1,...,*n*-1 is *circulant* if the
permutation (0,1,...,*n*-1) is an automorphism. In the following
gzipped tar files are text files with names of the form
*n*>.<*d*>.txt*n* vertices and degree *d*.
Each graph is given on one line as a set *S* of *d* integers.
The vertices
are 0,1,...,*n*-1 and the edges are all pairs {*x*,*y*} where
*x*−*y* is in *S* modulo *n*.

All degrees (up to complement) are present up to 60 vertices, then degrees at most 20 up to 65 vertices, at most 16 up to 70 vertices and at most 12 up to 100 vertices.

circ5.tar.gz circ6.tar.gz circ7.tar.gz circ8.tar.gz circ9.tar.gz circ10.tar.gz circ11.tar.gz circ12.tar.gz circ13.tar.gz circ14.tar.gz circ15.tar.gz circ16.tar.gz circ17.tar.gz circ18.tar.gz circ19.tar.gz circ20.tar.gz circ21.tar.gz circ22.tar.gz circ23.tar.gz circ24.tar.gz circ25.tar.gz circ26.tar.gz circ27.tar.gz circ28.tar.gz circ29.tar.gz circ30.tar.gz circ31.tar.gz circ32.tar.gz circ33.tar.gz circ34.tar.gz circ35.tar.gz circ36.tar.gz circ37.tar.gz circ38.tar.gz circ39.tar.gz circ40.tar.gz circ41.tar.gz circ42.tar.gz circ43.tar.gz circ44.tar.gz circ45.tar.gz circ46.tar.gz circ47.tar.gz circ48.tar.gz circ49.tar.gz circ50.tar.gz circ51.tar.gz circ52.tar.gz circ53.tar.gz circ54.tar.gz circ55.tar.gz circ56.tar.gz circ57.tar.gz circ58.tar.gz circ59.tar.gz circ60.tar.gz circ61.tar.gz circ62.tar.gz circ63.tar.gz circ64.tar.gz circ65.tar.gz circ66.tar.gz circ67.tar.gz circ68.tar.gz circ69.tar.gz circ70.tar.gz circ71.tar.gz circ72.tar.gz circ73.tar.gz circ74.tar.gz circ75.tar.gz circ76.tar.gz circ77.tar.gz circ78.tar.gz circ79.tar.gz circ80.tar.gz circ81.tar.gz circ82.tar.gz circ83.tar.gz circ84.tar.gz circ85.tar.gz circ86.tar.gz circ87.tar.gz circ88.tar.gz circ89.tar.gz circ90.tar.gz circ91.tar.gz circ92.tar.gz circ93.tar.gz circ94.tar.gz circ95.tar.gz circ96.tar.gz circ97.tar.gz circ98.tar.gz circ99.tar.gz circ100.tar.gz

On the extremal page we give a collection of graphs which have the greatest possible number of edges given the the number of vertices and the absence of some specified cycle lengths.

Page Master: Brendan McKay, brendan.mckay@anu.edu.au and https://users.cecs.anu.edu.au/~bdm.