Normalized solutions of $L^2$-supercritical NLS equations on noncompact metric graph with vanishing potential and localized nonlinearities

We study the existence of normalized solutions to the $L^2$-supercritical nonlinear Schr\"odinger equation on a noncompact metric graph $G$, $$\begin{cases} -u''+W(x)u+\lambda u=\chi(x)|u|^{p-2}u, & \text{on every edge } e \text{ of } G,\\[2mm] \displaystyle\sum_{e\succ v}u'_e(v)=0, & \text{at every vertex } v\in V, \end{cases}$$ under the mass constraint $\int_G |u|^2\,dx=\mu>0$, where $\lambda$ arises as a Lagrange multiplier. Here $p>6$, the potential $W$ belongs to $L^\infty(G)$, is nonnegative and vanishes at infinity along every unbounded edge of $G$, and $\chi$ is the characteristic function of the compact core $\mathcal{K}$, so that the nonlinearity is localized. In this regime, the energy functional is unbounded from below on the mass constraint, and, since metric graphs are not scale invariant, the scaling arguments and the Pohozaev identity available in the Euclidean setting cannot be used. We prove that, for every $\mu>0$, the problem admits a positive solution with $\lambda>0$, arising as a constrained critical point at a strictly positive energy level. The proof combines a uniform mountain-pass geometry for a family of approximating functionals, the monotonicity trick together with Morse-type information on the associated Palais-Smale sequences, and a blow-up analysis which rules out the divergence of the Lagrange multipliers. To the best of our knowledge, this is the first existence result for normalized solutions of the $L^2$-supercritical NLS equation on a noncompact metric graph in the presence of an external potential.