On the Independence Polynomial and Threshold of an Antiregular \(k\)-Hypergraph

Given an integer \(k\geq 3\) and an initial \(k-1\) isolated vertices, an {\em antiregular \(k\)-hypergraph} is constructed by alternatively adding an isolated vertex (connected to no other vertices) or a dominating vertex (connected to every other \(k-1\) vertices). Let \(a_i\) be the number of independent sets of cardinality \(i\) in a hypergraph \(H\), then the {\em independence polynomial} of \(H\) is defined as \(I(H;x)=\sum_{i=0}^m a_i x^i\), where \(m\) is the size of a maximum independent set. The main purpose of the present paper is to generalise some results of independence polynomials of antiregular graphs to the case of antiregular \(k\)-hypergraphs. In particular, we derive (semi-)closed formulas for the independence polynomials of antiregular \(k\)-hypergraphs and prove their log-concavity. Furthermore, we show that antiregular \(k\)-hypergraphs are {\em \(T2\)-threshold}, which means there exist a labeling \(c\) of the vertex set and a threshold \(\tau\) such that for any vertex subset \(S\) of cardinality \(k\), \(\sum_{i\in S}c(i)>\tau\) if and only if \(S\) is a hyperedge.