Ben, all the edges are included to form the angles, sorry for unclearness. The smallest angle obtained using the centre would be an interior angle of the triangle that contains the centre, a vertex on the boundary and its farthest vertex on the boundary, so it can be much less than $\pi/2m$.

Squark, $v_{m+1}$ is also a vertex in the whole graph, sorry for unclearness. In an equilateral polygon, the smallest angle would be an interior angle of the triangle that has $v_{m+1}$ as a vertex, so it would be less than the one you expressed.