On Symmetry of Independence Polynomials

Abstract: 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 size k in G, then I(G; x) = s₀ + s₁x + s₂x₂ + ... + s_αx^α, α = α (G), is called the independence polynomial of G (Gutman and Harary, 1986). If sj = s_αj for all 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 (Stevanović, 1998). In this paper we develop a new algebraic technique in order to take care of symmetric independence polynomials. On the one hand, it provides us with alternative proofs for some previously known results. On the other hand, this technique allows to 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).