  In this paper we show that absence of complex zeros of the partition function
of the hard-core model on any family of bounded degree graphs implies that the
associated probability measure, the \emph{hard-core measure}, satisfies strong
spatial mixing on that family. As a corollary we obtain that the hard-core
measure on the family of bounded degree claw-free graphs satisfies strong
spatial mixing. We furthermore derive strong spatial mixing for graph
homomorphism measures from absence of zeros of the graph homomorphism partition
function.
