On the independence polynomial of the corona of graphs

Let α(G) be the cardinality of a largest independent set in graph G. If _sk is the number of independent sets of size k in G, then I(G;x)=_s0+_s1x+⋯+_sα^xα, α=α(G), is the independence polynomial of G (Gutman and Harary, 1983). I(G;x) is palindromic if _sα-i=_si for each iε{0,1,⋯,⌊α/2⌋}. The corona of G and H is the graph G⊙H obtained by joining each vertex of G to all the vertices of a copy of H (Frucht and Harary, 1970). In this paper, we show that I(G⊙H;x) is palindromic for every graph G if and only if H=_Kr-e,r≥2. In addition, we connect realrootness of I(G⊙H;x) with the same property of both I(G;x) and I(H;x).