A simple proof of an inequality connecting the alternating number of independent sets and the decycling number

If s k denotes the number of independent sets of cardinality k and α ( G ) is the size of a maximum independent set in graph G , then I ( G ; x ) = s 0 + s 1 x + ⋯ + s α ( G ) x α ( G ) is the independence polynomial of G (Gutman and Harary, 1983) [8]. s paper we provide an elementary proof of the inequality | I ( G ; − 1 ) | ≤ 2 φ ( G ) (Engström, 2009) [7], where φ ( G ) is the decycling number of G (Beineke and Vandell, 1997) [3], namely, the minimum number of vertices that have to be deleted in order to turn G into a forest.