The Combinatorial Object Server

Number $n$ of vertices | (max. 20) | Connected | |

Minimum number of edges | Biconnected | ||

Maximum number of edges | Triangle-free | ||

Minimum degree $\geq$ | 4-cycle-free | ||

Maximum degree $\leq$ | Bipartite | ||

Canonic labeling | |||

Graph format | |||

Output format | |||

Output
numbering
graphics

Graphs are among the most heavily studied objects in combinatorics.
However, despite their simple definition, they are much more complex than the basic objects like permutations, combinations, partitions etc.
Formally, a graph is a pair $G=(V,E)$, where $V$ is a finite nonempty set and $E$ is a set of unordered distinct pairs of elements from $V$, called the of vertices and edges of the graph, respectively.
To emphasize that loops and multiple edges are disallowed, such graph are sometimes called simple.
Here we consider graphs to be unlabelled, which means we consider them the same if they differ only in renaming of the vertices.
We write $n:=|V|$ for the number of vertices, and we always consider the vertex set $V=\{1,2,\ldots,n\}$ for simplicity.
Graphs can be represented in a computer in a variety of ways.
An edge list is a list of all edges of the graph.
Another representation is the adjacency list, which is a list that specifies for each vertex $i$ the list of all its neighbors, i.e., all edges $ij\in E$.
A graph can also be represented by its adjacency matrix, which is an $n\times n$ matrix $A$ with $a_{i,j}=1$ if $ij\in E$ and $a_{i,j}=0$ otherwise.
As we only consider undirected graphs, we always have $a_{i,j}=a_{j,i}$.
For example, consider the graph shown below.

Its edge list representation is 1,2; 1,3; 2,3; 2,4; 3,4, its adjacency list representation is 1[2 3] 2[1 3 4] 3[1 2 4] 4[2 3], and the adjacency matrix representation is 0110 1011 1101 0110 (written out row by row).

OEIS sequences related to unlabelled $n$-vertex graphs:

- All graphs: OEIS A000088
- Triangle-free graphs: OEIS A006785
- 4-cycle-free graphs: OEIS A006786
- Triangle- and 4-cycle-free graphs: OEIS A006787
- Bipartite graphs: OEIS A033995
- 4-cycle-free bipartite graphs: OEIS A138347
- Graphs with $n$ vertices and $n$ edges: OEIS A001434
- Connected graphs: OEIS A001349
- Connected triangle-free graphs: OEIS A024607
- Connected 4-cycle-free graphs: OEIS A077269
- Connected triangle- and 4-cycle-free graphs: OEIS A126757
- Connected bipartite graphs: OEIS A005142
- Connected 4-cycle-free bipartite graphs: OEIS A300749
- Connected graphs with $n$ vertices and $n$ edges: OEIS A001429
- Unlabelled trees: OEIS A000055

- [MP14] B. D. McKay and A. Piperno. Practical graph isomorphism, II. J. Symbolic Comput., 60:94–112, 2014.