Trees with non log-concave independent set sequences

Abstract

We construct a family of trees with independence numbers going to infinity for which the log-concavity relation for the independent set sequence of a tree $T$ in the family fails at around $\hat{I}\pm (T)\left(1-1/(16\log \hat{I}\pm (T))\right)$. Here $\hat{I}\pm (T)$ is the independence number of $T$. This resolves a conjecture of Kadrawi and Levit.

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