The independence fractal of a graph

Independence polynomials of graphs enjoy the property of essentially being closed under graph composition (or ‘lexicographic product’). We ask here: for higher products of a graph G with itself, where are the roots of their independence polynomials approaching? We prove that in fact they converge (in the Hausdorff topology) to the Julia set of the independence polynomial of G, thereby associating with G a fractal. The question arises as to when these fractals are connected, and for graphs with independence number 2 we exploit the Mandelbröt set to answer the question completely.