  We prove that the eigenfunctions of quantum star graphs exhibit multifractal
self-similar structure in certain specified circumstances. In the semiclassical
regime, when the spectral parameter and the number of vertices tend to
infinity, we derive an asymptotic condition for the Mellin transform of a
specified function arising from the set of bond lengths which yields an
asymptotic for the Renyi entropy associated with an eigenfunction. We apply
this result to show that one may construct simple quantum star graphs which
satisfy a multifractal scaling law. In the low frequency regime we prove
multifractality by computing the Renyi entropy in terms of a zeta function
associated with the set of bond lengths. In certain arithmetic cases the
fractal exponent D_q satisfies a symmetry relation around q=1/4 which arises
from the functional equation of the zeta function. Our results are, in some
sense, analogous to the multifractal scaling law that the authors recently
proved for arithmetic Seba billiards. However, unlike in that case, we do not
require arithmetic conditions to be satisfied, and nor do we rely on delicate
arithmetic estimates.
