On the independent set sequence of a tree

TL;DR

This paper investigates the behavior of the independent set sequence in trees, showing asymptotic properties for random trees and extending results to K"onig-Egerváry graphs, using combinatorial and computational methods.

Contribution

It provides asymptotic analysis of the independent set sequence in random trees and generalizes existing results to a broader class of graphs.

Findings

Approximately 49.5% of the sequence is increasing in random trees

Approximately 38.8% of the sequence is decreasing in random trees

Extended results on the final third of the independent set sequence for K"onig-Egerváry graphs

Abstract

Alavi, Malde, Schwenk and Erd\H{o}s asked whether the independent set sequence of every tree is unimodal. Here we make some observations about this question. We show that for the uniformly random (labelled) tree, asymptotically almost surely (a.a.s.) the initial approximately 49.5\% of the sequence is increasing while the terminal approximately 38.8\% is decreasing. Our approach uses the Matrix Tree Theorem, combined with computation. We also present a generalization of a result of Levit and Mandrescu, concerning the final one-third of the independent set sequence of a K\"onig-Egerv\'ary graph.