# Labeled graph

Labeled graph

In graph theory (which is an area in mathematics and computer science) a "labeled graph" is a graph with "labels" assigned to its nodes and edges. These assignments do not have to be unique, i.e. different nodes or edges can have the same label. Mathematically a labeled graph can be defined as follows:

Given two alphabets $Sigma_V$ and $Sigma_E$ a labeled graph is a triple $G = \left(V,E,ell_V\right)$ where
*$V$ is a finite set of nodes,
*$Esubseteq V imesSigma_E imes V$ is a ternary relation describing the edges (including the labeling of the edges) and
*$ell_Vcolon V ightarrowSigma_V$ is a function describing the labeling of the nodes.If E is symmetric, i.e. for all edges $\left(v,l,w\right)in E$ it is also $\left(w,l,v\right)in E$, then the Graph G is called undirected and directed otherwise.

Note that this notion of a labeled graph is different from the notion given in the article graph labeling.

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