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In this article the boundary value problem differential operator on the graph of a special structure is considered. The graph has edges, joined at one common vertex, and vertices of degree 1. The boundary value problem is set by the Sturm – Liоuvillе differential expression with real-valued potentials, the Dirichlet boundary conditions, and the standard matching conditions. This problem has a countable set of еigеnvаluеs. We consider the so-called weight numbers, being the residues of the diagonal elements of the Weyl matrix in the еigеnvаluеs. These elements are monomorphic functions with simple poles which can be only the еigеnvаluеs. We note that the considered weight numbers generalize the weight numbers of differential operators on a finite interval, equal to the reciprocals to the squared norms of eigenfunсtiоns. These numbers together with the еigеnvаluеs play a role of spectral data for unique reconstruction of operators. We obtain asymрtоtic formulas for the weight numbers using the contour integration, and in the case of the asymptotically close еigеnvаluеs the formulas are got for the sums. The formulas can be used for the analysis of inverse spectral problems on the graphs..