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We consider a compact metric graph of size $\varepsilon$, and attach to itseveral edges (leads) of length of order one (or of infinite length). As$\varepsilon$ goes to zero, the graph $\mathcal{G}^\varepsilon$ obtained inthis way looks like the star-graph formed by the leads joined in a centralvertex. On $\mathcal{G}^\varepsilon$ we define an Hamiltonian $H^\varepsilon$,properly scaled with the parameter $\varepsilon$. We prove that there exists ascale invariant effective Hamiltonian on the star-graph that approximates$H^\varepsilon$ (in a suitable norm resolvent sense) as $\varepsilon\to0$. Theeffective Hamiltonian depends on the spectral properties of an auxiliary$\varepsilon$-independent Hamiltonian defined on the compact graph obtained bysetting $\varepsilon = 1$. If zero is not an eigenvalue of the auxiliaryHamiltonian, in the limit $\varepsilon\to0$, the leads are decoupled.

Scale Invariant Effective Hamiltonians for a Graph with a Small Compact Core