Directed Graph-weight

[mathmari]

Hii!!!

I hope someone can help me with the following exercise...!!!

Let G=(V,E) be a directed graph with weight w: E->R, n=|V|.

Let m( c )=1/k*sum(w(e_{i}), i=1,k) be the mean weight of a circle.

Let m*=min(m( c )), the minimum of the mean weights of the circles of G.

Is it correct to say that, since the minimum of the mean weight is 0 and not negative, there are no circles with negative weight? Or is there an other explanation that if m*=0 there are no circles with negative weight..???

And also, how could I explain that δ(s,v)=min(δ_{k}(s,v), o<=k<=n-1), where δ(s,v) is the weight of the lightest path from s to v, and δ_{k}(s,v) is the weight of the lightest path from s to v that contains exactly k vertices (when there is no path from s to v with k vertices δ_{k}(s,v)=infinity)??