ACTA UNIVERSITATIS APULENSIS Special Issue GRAPHS WITH F -SYMMETRIC INDEPENDENCE POLYNOMIALS

An independent set in a graph is a set of pairwise non-adjacent vertices, and α(G) is the size of a maximum independent set in the graph G. If sk is the number of independent sets of cardinality k in G, then I(G;x) = s0 + s1x+ s2x 2 + ...+ sαx , α = α (G) , is called the independence polynomial of G (I. Gutman and F. Harary, 1983). If sα−i = f (i) · sα−j holds for every i ∈ {0, 1, ..., bα/2c}, then I(G;x) is called f -symmetric. The corona of the graphs G and H is the graph G ◦ H obtained by joining each vertex of G to all the vertices of a copy of H. In this paper we show that for every graph G, the independence polynomial of G ◦ (Kp ∪Kq) is f -symmetric, where f (i) = (pq) α 2 −i , 0 ≤ i ≤ ⌊ α 2 ⌋ , α = α (G ◦ (Kp ∪Kq)) . In particular, we deduce a result of Stevanović [20], claiming that I (G ◦ 2K1;x) is symmetric, i.e., sα−i = sα−j holds for every i ∈ {0, 1, ..., bα (G ◦ 2K1) /2c}.