<p>We study the bistable reaction–diffusion equation on metric graphs subject to Neumann boundary conditions at the endpoints and Kirchhoff’s law at the nodes. Specifically, we focus on star-graphs and graphs constructed by joining multiple star-graphs. The purpose of this article is to construct stable and unstable nonconstant stationary solutions with precise profiles for these graphs with sufficiently long edges. To achieve this, we derive a reduced energy functional for approximate solutions on the star-graph and identify the solutions and their stability in terms of critical points of this energy. By gluing together solutions obtained for identical star-graphs, we extend them to the graphs formed by joining the star-graphs. We then apply the comparison principle to establish the stability of the solutions. A variety of graphs allowing stable and unstable nonconstant stationary solutions are also demonstrated as concrete examples.</p>