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

If ^sk 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)=^s0+^s1x+⋯+sα( _G)xα(^G) is the independence polynomial of G (Gutman and Harary, 1983) [8]. In this paper we provide an elementary proof of the inequality|I(G;-1)|≤2^φ(G) (Engstrm, 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.