2604.01185 The Independence Polynomial Root Density: Zeros of Independence Polynomials of Grid Graphs Concentrate on a Cardioid Curve
We compute independence polynomials I(G,x) for grid graphs G_{m,n} with m,n <= 20 and analyze the distribution of their complex roots. For fixed strip width m and increasing length n, we prove that the roots of I(G_{m,n}, x) converge to an algebraic curve in the complex plane that is a cardioid whose parametric equation depends on the spectral radius of the transfer matrix for independent sets on the m-wide strip.