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Partial differential equation on networks have been widely investigated inthe last decades in view of their application to quantum mechanics(Schr\"odinger type equations) or to the analysis of flexible structures (wavetype equations). Nevertheless, very few results are available for diffusivemodels despite an increasing demand arising from life sciences such asneurobiology. This paper analyzes the controllability properties of the heatequation on a compact network under the action of a single input bilinearcontrol.  By adapting a recent method due to [F.~Alabau-Boussouira, P.~Cannarsa andC.~Urbani, {\em  Exact controllability to eigensolutions for evolution equations of parabolictype via bilinear control}, arXiv:1811.08806], an exact controllability resultto the eigensolutions of the uncontrolled problem is obtained in this work. Acrucial step has been the construction of a suitable biorthogonal family undera non-uniform gap condition of the eigenvalues of the Laplacian on a graph.Application to star graphs and tadpole graphs are included.

Exact controllability to eigensolutions of the bilinear heat equation on compact networks