used these codes to might a simple triangle, which nodes have different effects each other:

data = {{0, 1, 0}, {0, 0, 2}, {3, 0, 0}}; AdjacencyGraph[data, VertexLabels -> "Name"] 

thats gave me between nodes 1 and 3 three edges , futhermore between nodes 2 and 3 two edges.
How can I show on a graph from thin to thickness these values (1,2,3)

@prgrad and @m_goldberg

I might to transform a graph which gives me single edge between the nodes. See pic.:

for each value of other different. for example +3 dark black, for -3 dark red etc.

you can maybe help me futher with these (minus and plus) codes:

data = {{0, -1, 0}, {0, 0, 2}, {3, 0, 0}}; AdjacencyGraph[data, VertexLabels -> "Name"]