We show that there are no stable stationary nonconstant solutions of the evolution problem (1) for fully autonomous reaction-diffusion-equations on the edges of a finite metric graph $ G$ under continuity and Kirchhoff flow transition conditions at the vertices.
$(1) \ \ \ \ \ \ \ \ \ \ \begin{cases} u∈ \mathcal{C}(G×[0,∞))\cap \mathcal{C}^{2,1}_{K}(G×(0,∞)),\\\partial_t u_j=\partial_j^2u_{j}+f(u_j) & \text{on the edges }k_j,\\ \displaystyle(K)\ \ \ \ \sum\limits_{j=1}^N d_{ij} c_{ij}\partial_ju_{j}(v_i,t)=0 &\text{at the vertices } v_i.\end{cases} $
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