<p>When a graph has parallel edges and self loops one cannot identify an edge with the set of two adjacent vertices it connects.  I have failed to find a good formal definition of a non-simple graph.  This is my attempt: </p>

<blockquote>
  <p>An edge weighted graph $G = (V, E, \omega, \gamma)$ is a set of vertices $V$ together with a set of edge $E$, a function $\gamma: E \rightarrow \{\{u, v\}: u, v \in V\}$ and a weight function $\omega: E\rightarrow R$.</p>
</blockquote>

<p>I am wondering whether this is sound, and what is a good standard name for the function $\gamma$ in this context.  </p>

CC BY-SA 3.0
2015-04-20T17:35:18.157Z
No, it should not be defined as a multiset.  Otherwise one cannot assign different weights to two edges with the same endpoints.
George Octavian Rabanca
CC BY-SA 3.0
2015-04-20T17:35:32.777Z
The definition looks correct, and except for weights, it’s identical to the one given in https://en.wikipedia.org/wiki/Multigraph#Undirected_multigraph_.28edges_with_own_identity.29 .
Emil Jeřábek
CC BY-SA 3.0
2015-04-21T15:37:16.543Z
Thank you.  I guess my main question is whether there is a standard name for the function $\gamma$.  I wouldn't want to confuse the readers by make up a new name for something standard.
George Octavian Rabanca
<graph-theory>
Proper definition of a graph with with loops and parallel edges
George Octavian Rabanca
# Proper definition of a graph with with loops and parallel edges

(Asked by: username_0 on 2015-04-20)

When a graph has parallel edges and self loops one cannot identify an edge with the set of two adjacent vertices it connects. I have failed to find a good formal definition of a non-simple graph. This is my attempt:

> An edge weighted graph $G = (V, E, \omega, \gamma)$ is a set of vertices $V$ together with a set of edge $E$, a function $\gamma: E \rightarrow \{\{u, v\}: u, v \in V\}$ and a weight function $\omega: E\rightarrow R$.

I am wondering whether this is sound, and what is a good standard name for the function $\gamma$ in this context.

**Tags:** graph-theory

> **Comments:**
> **username_0**: No, it should not be defined as a multiset.  Otherwise one cannot assign different weights to two edges with the same endpoints.  
> **username_1**: The definition looks correct, and except for weights, it’s identical to the one given in https://en.wikipedia.org/wiki/Multigraph#Undirected_multigraph_.28edges_with_own_identity.29 .  
> **username_0**: Thank you.  I guess my main question is whether there is a standard name for the function $\gamma$.  I wouldn't want to confuse the readers by make up a new name for something standard.  

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## 0 answers

