The independence fractal of a graph. (Q1405113)

The authors consider the roots of the independence polynomials of higher (lexicographic) products of a graph with itself. The main result of the paper (Theorem 3.3) describes where the roots of the (reduced) independence polynomial for \(G^k\) are approaching. In particular, an association of a fractal with \(G\) is shown which is given by the limit of the sequence of \(f_{G^k}\) for \(k \rightarrow \infty\). The so defined independence fractal of \(G\) is shown to be the Julia set of its reduced independence polynomial \(f_{G}(x)\). Furthermore, the connectedness of the independence fractal is studied and for graphs with independence number 2 the {Mandelbrot set} is considered.