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What conceptually are negative weights on graphs? Why might they have them?

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What is the significance of negative weight edges in a graph?

Above link answers your query.

One more example:- If we want to design some architecture then in the whole circuitry there is flow of current. The whole circuitry can be modeled as graph with device components as vertices and the path between them as edges. In these paths the resistance can be taken as negative weights.

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