= Rank of the fundamental group of a finite connected graph
{title2=$\operatorname{rank}\pi_1(X)=e-v+1=1-\chi(X)$}
For a finite <connected graph>, collapse a <spanning tree>. Its $v-1$ <edges> disappear, and each of the remaining $e-v+1$ <edges> gives a <circle> in a <bouquet of circles>. The <fundamental group of a bouquet of circles> is <free group>, with one member of a <free basis of a group> for each <circle>. This gives the formula in terms of the <Euler characteristic> $\chi(X)=v-e$.
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