On the independence polynomial of an antiregular graph

A graph with at most two vertices of the same degree is called antiregular [25], maximally nonregular [32] or quasiperfect [2]. If sk is the number of independent sets of cardinality k in a graph G, then I(G;x) = s0 + s1x+ ...+ sαx α is the independence polynomial of G [10], where α = α(G) is the size of a maximum independent set. In this paper we derive closed formulae for the independence polynomials of antiregular graphs. In particular, we deduce that every antiregular graph A is uniquely defined by its independence polynomial I(A;x), within the family of threshold graphs. Moreover, I(A;x) is log-concave with at most two real roots, and I(A;−1) ∈ {−1, 0}.