  We provide a purely variational proof of the existence of eigenvalues below
the bottom of the essential spectrum for the Schr\"odinger operator with an
attractive $\delta$-potential supported by a star graph, i.e. by a finite union
of rays emanating from the same point. In contrast to the previous works, the
construction is valid without any additional assumption on the number or the
relative position of the rays. The approach is used to obtain an upper bound
for the lowest eigenvalue.
