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A well-known theorem of Wolpert shows that the Weil-Petersson symplectic formon Teichm\"uller space, computed on two infinitesimal twists along simpleclosed geodesics on a fixed hyperbolic surface, equals the sum of the cosinesof the intersection angles. We define an infinitesimal deformation startingfrom a more general object, namely a balanced geodesic graph, by which anytangent vector to Teichm\"uller space can be represented. We then prove ageneralization of Wolpert's formula for these deformations. In the case ofsimple closed curves, we recover the theorem of Wolpert.

Generalization of a formula of Wolpert for balanced geodesic graphs on closed hyperbolic surfaces