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In this paper, motivated by a biodegradable stent problem, we consider a parabolic differential equation on graphs. This kind of equation describes the stent material degradation in time. Since stents are mesh-like structures with thin struts, they can be modelled using a simple structure called the one-dimensional curved rod model. In this way, we obtain a graph-like domain for our parabolic equation. Here, we prove the regularity estimate for the unique solution of the equation, together with corresponding estimates. [ABSTRACT FROM AUTHOR]