In this talk I will introduce certain identities for independence polynomials, that are similar to the Christoffel--Darboux identities used in the theory of orthogonal polynomials. As an application, we will see that some conditions on the induced subgraphs of a graph G translates to results on zero-free regions of the independence polynomial of G. In particular we give a new proof of a theorem of Chudnovsky and Seymour, which states that the independence polynomial of a claw-free graph has only real zeros.