  We revisit classical bounds of M. E. Fisher on the ferromagnetic Ising model,
and show how to efficiently use them on an arbitrary given graph to rigorously
upper-bound the partition function, magnetizations, and correlations. The
results are valid on any finite graph, with arbitrary topology and arbitrary
positive couplings and fields. Our results are based on high temperature
expansions of the aforementioned quantities, and are expressed in terms of two
related linear operators: the non-backtracking operator and the Bethe Hessian.
As a by-product, we show that in a well-defined high-temperature region, the
susceptibility propagation algorithm converges and provides an upper bound on
the true spin-spin correlations.
