On Unimodality of Independence Polynomials of Some Well-Covered Trees

Levit, Vadim E.; Mândrescu, Eugen

doi:10.1007/3-540-45066-1_19

Cited by 20 publications

(18 citation statements)

References 13 publications

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“…Alavi, Maldi, Schwenk, and Erdös conjectured that the independence polynomial of every tree is unimodal [2], and Stanley conjectured that the chromatic symmetric function is a complete graph isomorphism invariant for trees [44]. In [27,28] and [32] the authors, respectively, verify that these conjectures hold for caterpillars and (some) spiders. We show in the following that these important families of graphs also yield nice properties for the generating polynomial M (G; x).…”

Section: Classic Families Of Treesmentioning

confidence: 97%

Counting Markov equivalence classes for DAG models on trees

Radhakrishnan

Solus

Uhler

2018

Discrete Applied Mathematics

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DAG models are statistical models satisfying a collection of conditional independence relations encoded by the nonedges of a directed acyclic graph (DAG) G. Such models are used to model complex cause-effect systems across a variety of research fields. From observational data alone, a DAG model G is only recoverable up to Markov equivalence. Combinatorially, two DAGs are Markov equivalent if and only if they have the same underlying undirected graph (i.e. skeleton) and the same set of the induced subDAGs i → j ← k, known as immoralities. Hence it is of interest to study the number and size of Markov equivalence classes (MECs). In a recent paper, the authors introduced a pair of generating functions that enumerate the number of MECs on a fixed skeleton by number of immoralities and by class size, and they studied the complexity of computing these functions. In this paper, we lay the foundation for studying these generating functions by analyzing their structure for trees and other closely related graphs. We describe these polynomials for some important families of graphs including paths, stars, cycles, spider graphs, caterpillars, and complete binary trees. In doing so, we recover important connections to independence polynomials, and extend some classical identities that hold for Fibonacci numbers. We also provide tight lower and upper bounds for the number and size of MECs on any tree. Finally, we use computational methods to show that the number and distribution of high degree nodes in a triangle-free graph dictates the number and size of MECs.

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Section: Classic Families Of Treesmentioning

confidence: 97%

Counting Markov equivalence classes for DAG models on trees

Radhakrishnan

Solus

Uhler

2018

Discrete Applied Mathematics

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“…By the assumption k 0.49n, and the fact that t k + 1, we have that the endpoints of the interval in (21) are positive, with the lower endpoint bounded away from 0 and the upper endpoint bounded away from infinity. Also note that f (z) has power series about 0 with all coefficients positive, and with infinite radius of convergence.…”

Section: Deriving (5)mentioning

confidence: 99%

“…There have been numerous partial results, mostly exhibiting families of trees with unimodal independent set sequences, see e.g. [2,5,13,21,22,23,29,32,33,34,35]. The unimodality of the independent set sequence of all forests on at most 25 vertices has been verified computationally [27,31], but the full question remains stubbornly open.…”

Section: Introductionmentioning

confidence: 99%

On the Independent Set Sequence of a Tree

Basit

Galvin

2021

Electron. J. Combin.

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Alavi, Malde, Schwenk and Erdős 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 König-Egerváry graph.

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“…In [17] and [18], the unimodality of independence polynomials of a number of well-covered trees (e.g., P * n , K * 1,n ) is validated, using the result, mentioned above, on claw-free graphs due to Hamidoune, or directly, by identifying the location of the mode. These findings seem promising for proving Conjecture 1.2 in the case of very well-covered trees, since a tree T is well-covered if and only if either T is a well-covered spider (i.e., T ∈ {K 1 , K * 1 , K * 1,n : n ≥ 1}), or T is obtained from a well-covered tree H 1 and a well-covered spider H 2 , by adding an edge joining two non-pendant vertices belonging to H 1 , H 2 , respectively (see [16]).…”

Section: Introductionmentioning

confidence: 99%