On Symmetry of 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. A matching is a set of non-incident edges, while μ(G) is the cardinality of a maximum matching. 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 (Gutman and Harary [7]). If sj = sα−j , 0 ≤ j ≤ ⌊α/2⌋, then I(G;x) is called symmetric (or palindromic). It is known that the graph G ◦ 2K1 obtained by joining each vertex of G to two new vertices, has a symmetric independence polynomial [23]. In this paper we show that for every graph G and for each non-negative integer k ≤ μ (G), one can build a graph H , such that: G is a subgraph of H , I (H ;x) is symmetric, and I (G ◦ 2K1;x) = (1 + x) k · I (H ;x).