Definition:Simple Graph/Formal Definition

Definition

Let $V$ be a set.

Let $\RR$ be an endorelation on $V$ which is antireflexive and symmetric.

Let $E$ be the set whose elements of the form:

$\set {\tuple {v_a, v_b}, \tuple {v_b, v_a} }$.

where $\tuple {v_a, v_b}$ and $\tuple {v_b, v_a}$ are elements of $\RR$

A simple graph is an ordered pair $G = \struct {V, E}$, where $V$ and $E$ are defined as above.

$V$ is called the vertex set.

$E$ is called the edge set.

Also presented as

It is usually more convenient to express the elements of the edge set as doubletons, of the form $\set {v_a, v_b}$.

Also defined as

Some sources impose the condition that $V \ne \O$.

Some sources also define a (simple) graph as one such that $V$ is a finite set.

Examples

Arbitrary Order $4$ Graph

Let $V = \set {v_1, v_2, v_3, v_4}$.

Let $\RR = \set {\tuple {v_1, v_2}, \tuple {v_1, v_3}, \tuple {v_2, v_1}, \tuple {v_2, v_3}, \tuple {v_3, v_1}, \tuple {v_3, v_2}, \tuple {v_3, v_4}, \tuple {v_4, v_3} }$.

Then:

$E = \set {\set {\tuple {v_1, v_2}, \tuple {v_2, v_1} }, \set {\tuple {v_1, v_3}, \tuple {v_3, v_1} }, \set {\tuple {v_2, v_3}, \tuple {v_3, v_2} }, \set {\tuple {v_3, v_4}, \tuple {v_4, v_3} } }$

Arbitrary Order $5$ Graph

Let $G = \struct {V, E}$ be a simple graph such that:

$V = \set {v_1, v_2, v_3, v_4, v_5}$
$E = \set {v_1 v_2, v_1 v_4, v_1 v_5, v_2 v_3, v_3 v_5, v_4 v_5}$

Then $G$ can be presented in diagram form as:

The underlying relation $\RR$ on $V$ which defines the edge set of $G$ is:

$\RR = \set {\tuple {v_1, v_2}, \tuple {v_2, v_1}, \tuple {v_1, v_4}, \tuple {v_4, v_1}, \tuple {v_1, v_5}, \tuple {v_5, v_1}, \tuple {v_2, v_3}, \tuple {v_3, v_2}, \tuple {v_3, v_5}, \tuple {v_5, v_3}, \tuple {v_4, v_5}, \tuple {v_5, v_4} }$