Unimodality of some independence polynomials via their palindromicity

An independent set in a graph G is a set of pairwise non-adjacent vertices, and the independence number α(G) is the cardinality of a maximum independent set. The independence polynomial of G is I(G; x) = s0 + s1x+ s2x 2 + ...+ sαx , α = α(G), where sk equals the number of independent sets of size k in G (Gutman and Harary, 1983). If si = sα−i, 0 ≤ i ≤ α/2 , then I(G; x) is called palindromic. It is known that the graph G ◦ 2K1, obtained by joining each vertex of G to two new vertices, has a palindromic independence polynomial (Stevanović, 1998). In this paper we show that for every graph G, the polynomial I(G ◦ 2K1; x) is also unimodal.