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# 1. Definition & Example

A (finite, undirected, simple)

**graph**$G = (V,E)$ consists of a finite set $V$ of vertices and a set $E \subseteq \binom{V}{2}$ of edges.Two graphs $G = (V,E)$ and $G = (V',E')$ are

**isomorphic**if there is a bijection $\psi : V \rightarrow V'$ such that$$\{u,v\} \in E \Leftrightarrow \{\psi(u),\psi(v)\} \in E'.$$

**Unlabelled graphs**on $n$ vertices are graphs up to graph-isomorphism.A

**canonical form**of an unlabelled graph is a relabelling of the vertices (an isomorphic graph) such that any two graphs are isomorphic if and only if they have the same canonical form.

the 4 Graphs of size 3 | |||

([],3) |
([(1,2)],3) |
([(0,2),(1,2)],3) |
([(0,1),(0,2),(1,2)],3) |

- Unlabelled graphs are graphically represented by their unlabelled vertices with edges connecting adjacent vertices.
For the number of unlabelled graphs see OEIS:A000088.

# 2. Further definitions

If $\{u,v\}$ is an edge in a graph $G$, then $u$ and $v$ are

**adjacent vertices**. $u$ and $v$ are also known as**neighbors**. The set of neighbors of $v$, denoted $N(v)$, is called the**neighborhood**of $v$. The**closed neighborhood**of $v$ is $N[v]=N(v)\cup {v}$.If two edges share a vertex in common (e.g. $\{u,v\}$ and $\{v,w\})$, then they are

**adjacent edges**.The

**degree**of a vertex $v$, denoted**deg($v$)**, is the number of vertices adjacent to $v$.We call $|V(G)|$, the cardinality of the vertices of a graph $G$, the

**order**of the graph. We also say $|E(G)|$, the cardinality of the edges of a graph $G$, is the**size**of the graph.A graph of size 0 is called an

**empty graph**. Any graph with at least one edge is called**nonempty**.A graph is

**complete**when any two distinct vertices are adjacent. The complete graph of $n$ vertices is notated $K_{n}$A

**planar graph**is a graph that can be embedded in the plane. I.e. it can be drawn on the plane such a way that its edges intersect only at their endpoints.A

**walk**$W$ in a graph $G$ is a sequence of vertices in $G$, beginning at a vertex $u$ and ending at a vertex $v$ such that the consecutive vertices in $W$ are adjacent in $G$.A walk whose initial and terminal vertices are distinct is called an

**open walk**, otherwise it is a**closed walk**.A walk in which no edge repeats is called a

**trail**.A

**path**$P$ in a graph $G$ is a sequence of edges which connect a sequence of vertices which, are all distinct from one another. A path can also be thought of as a walk with no repeated vertex.A

**simple path**is one which contains no repeated vertices (in other words, it does not cross over itself).If there is a path from a vertex $u$ to a vertex $v$ then these two vertices are said to be

**connected**. If every two vertices in a graph $G$ are connected, then $G$ is itself a**connected graph.**A nontrivial closed walk in a graph $G$ in which no edge is repeated is a

**circuit**in $G$.A circuit with vertices $v_1, v_2, ..., v_k, v_1$ where $v_2, ..., v_k$ are all distinct is called a

**cycle**.Let $G$ be a nontrivial connected graph. A circuit $C$ of $G$ that contains every edge of $G$ (necessarily exactly once) is called an

**Eulerian Circuit**. Any graph which contains an Eularian Circuit is called**Eularian**. A graph is Eulerian if and only if all its vertices have even degrees.

A nontrivial connected graph $G$ is Eularian if and only if every vertex of G has even degree.

- Every planar graph is four-colorable. That is, the chromatic number of a planar graph is at most four.

# 3. References

G. Chartrand, L. Lesniak, and P. Zhang.

*Graphs and Digraphs.*CRC Press, Oct. 2010.

# 4. Sage examples

# 5. Technical information for database usage

A graph is uniquely represented as a tuple

`(E,n)`where`E`is the sorted list of edges in the canonical labelling and`n`is the number of vertices.- Graphs are graded by the number of vertices.
- The database contains all graphs of size at most 7.