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We study a topological space obtained from a graph via replacing verticeswith smooth Riemannian manifolds, i.e. a decorated graph. We construct asemiclassical asymptotics of the solutions of Cauchy problem for atime-dependent Schr\"{o}dinger equation on a decorated graph with a localizedinitial function. The main term of our asymptotic solution at an arbitraryfinite time is the sum of Gaussian packets and generalized Gaussian packets. Westudy the number of such packets as time goes to infinity. We prove asymptoticestimations for this number for the following decorated graphs: cylinder with asegment, two dimensional torus with a segment, three dimensional torus with asegment. Also we prove general theorem about a manifold with a segment andapply it to the case of a uniformly secure manifold.

Asymptotic estimate for the number of Gaussian packets on three decorated graphs