On the Independence Polynomial and Threshold of an Antiregular $k$-Hypergraph

Abstract

Given an integer $k\ge 3$ 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.