On the roots of independence polynomials

Abstract

The independence polynomial of a graph is a polynomial whose coefficients give the number of independent sets of each size. Its roots are called independence roots. This thesis explores the analytic properties of independence polynomials and the interactions between these properties and the structure of the corresponding graphs. We begin by applying results that relate the independence roots to the coefficients of independence polynomials of very well-covered graphs. We will explore families of graphs whose independence roots all lie to the left of the imaginary axis (which appears to be most graphs at a first glance) and other families of graphs that have independence roots to the right of this line. We then prove exponential bounds on the maximum modulus of an independence root that a graph of order $n$ can attain. Finally, we find graphs that are independence equivalent, that is have equivalent independence polynomial, to a path or a cycle of certain orders.