On the independence polynomial of the corona of graphs

Let α(G)α(G) be the cardinality of a largest independent set in graph GG. If sksk is the number of independent sets of size kk in GG, then I(G;x)=s0+s1x+⋯+sαxα, α=α(G)α=α(G), is the independence polynomial   of GG (Gutman and Harary, 1983). I(G;x)I(G;x) is palindromic   if sα−i=sisα−i=si for each i∈{0,1,…,⌊α/2⌋}i∈{0,1,…,⌊α/2⌋}. The corona   of GG and HH is the graph G∘HG∘H obtained by joining each vertex of GG to all the vertices of a copy of HH (Frucht and Harary, 1970).In this paper, we show that I(G∘H;x)I(G∘H;x) is palindromic for every graph GG if and only if H=Kr−e,r≥2H=Kr−e,r≥2. In addition, we connect realrootness of I(G∘H;x)I(G∘H;x) with the same property of both I(G;x)I(G;x) and I(H;x)I(H;x).