  We analyze the partition function of the Ising model on graphs of two
different types: complete graphs, wherein all nodes are mutually linked and
annealed scale-free networks for which the degree distribution decays as
$P(k)\sim k^{-\lambda}$. We are interested in zeros of the partition function
in the cases of complex temperature or complex external field (Fisher and
Lee-Yang zeros respectively). For the model on an annealed scale-free network,
we find an integral representation for the partition function which, in the
case $\lambda > 5$, reproduces the zeros for the Ising model on a complete
graph. For $3<\lambda < 5$ we derive the $\lambda$-dependent angle at which the
Fisher zeros impact onto the real temperature axis. This, in turn, gives access
to the $\lambda$-dependent universal values of the critical exponents and
critical amplitudes ratios. Our analysis of the Lee-Yang zeros reveals a
difference in their behaviour for the Ising model on a complete graph and on an
annealed scale-free network when $3<\lambda <5$. Whereas in the former case the
zeros are purely imaginary, they have a non zero real part in latter case, so
that the celebrated Lee-Yang circle theorem is violated.
