Rank of the fundamental group of a finite connected graph ($\mathrm{rank}{\pi }_1(X)=e-v+1=1-\chi (X)$)

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 $\chi (X)=v-e$.