On the independence polynomial of an antiregular graph

A graph with at most two vertices of the same degree is called antiregular (Merris 2003), maximally nonregular (Zykov 1990) or quasiperfect (Behzad, Chartrand 1967). If s_{k} is the number of independent sets of cardinality k in a graph G, then I(G;x) = s_{0} + s_{1}x + … + s_{alpha}x^{alpha} is the independence polynomial of G (Gutman, Harary 1983), where alpha = alpha(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 logconcave with at most two real roots, and I(A;-1) belongs to {-1,0}.

💡 Research Summary

This paper investigates the independence polynomial of antiregular graphs—graphs in which at most two vertices share the same degree. The authors first recall that antiregular graphs are also known as maximally non‑regular or quasiperfect graphs, and they belong to the family of threshold graphs. Using the recursive construction of threshold graphs (adding a new vertex either isolated or adjacent to all existing vertices), the paper derives explicit closed‑form expressions for the independence polynomial (I(A_n;x)) of an antiregular graph with (n) vertices.

For even order (n=2m) the polynomial is
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