# What is the definition of an weighted graph?

In graph theory which one of these two will be called a weighted graph ?

1. A graph where vertices have some weights or vales .
2. A graph where edges have some weights or values .
3. A graph where both edges and vertices have some weights or values
4. A graph where neither edges nor vertices have any weights or values

I couldn't find a picture for the third case actually .

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Note: The figure does not give examples of weighted graphs. These are labels and not weights. Labels may not be repeated, whereas weights may be. –  Douglas S. Stones Mar 26 '13 at 13:43

There is some variation in the literature, but typically a weighted graph refers to an edge-weighted graph, that is a graph where edges have weights or values. Without the qualification of weighted, the graph is typically assumed to be unweighted.

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Answer choice (2) according to one popular text:

With each edge $e$ of $G$ let there be associated a real number $w(e)$, called its weight. Then $G$, together with these weights on its edges, is called a weighted graph.$^1$

[1] Bondy and Murty. Graph Theory with Applications. North-Holland, 1976 (page 15-16)

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Weighted graph is a graph in which real number is associated with each edge of graph..

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A weighted graph is also directed graph with weight on the edge of each graph

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