Real-Rootedness via Pendant Extensions and Clique Graphs

The independent set sequences of graphs have been widely studied, but only recently have analogous questions been considered for strong independent set sequences of hypergraphs. We develop two tools to prove and preserve the real-rootedness of strong independence polynomials. The first is a pendant-edge transform for uniform pendant hyperedge attachments; it preserves real-rootedness when one or two pendant hyperedges are attached at each vertex, and this range is sharp. The second uses clique graphs and a theorem of Chudnovsky and Seymour on real-rootedness of independence polynomials of claw-free graphs to prove that any finite hypergraph in which every vertex lies in at most two hyperedges has a real-rooted strong independence polynomial. For linear hypertrees, this degree condition exactly characterizes when the clique graph is claw-free.

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