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Symmetric and unimodal independence polynomials of trees

T0 review · 0 major / 3 minor · reviewed 2026-05-10 · grok-4.3

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Pith's one-line read Trees on n vertices exist with symmetric and unimodal independence polynomials, and some symmetric unimodal polynomials of degree n arise from trees.

desk verdict The paper partially answers existence questions for trees whose independence polynomials are both symmetric and unimodal, mainly via constructions for small n and selected families. read the letter →

arxiv 2604.18824 v1 submitted 2026-04-20 math.CO

Takayuki Hibi , Selvi Kara , Dalena Vien This is my paper · ORCID
classification math.CO
keywords independencepolynomialtreessymmetricunimodalgraphpolynomialscombinatoricsindependentsets
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper examines whether, for a given n, there is a tree on exactly n vertices whose independence polynomial has coefficients that are symmetric, reading the same forwards and backwards, and unimodal, increasing to a peak and then decreasing. It separately checks whether any given symmetric and unimodal polynomial of degree n can be realized as the independence polynomial of some tree. A reader would care because the independence polynomial records how many independent sets of each size exist in the graph, so symmetry and unimodality reveal a balanced distribution of set sizes that may be special to tree structures.

What carries the argument

The independence polynomial of a tree, the generating function whose coefficient of x^k counts the independent sets of size k, with its coefficient sequence checked for symmetry and unimodality.

What would settle it

An explicit n where every tree on n vertices has an independence polynomial that is either non-symmetric or non-unimodal, or a symmetric unimodal polynomial of degree n that is not the independence polynomial of any tree.

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Extended reading notes

Core claim

We study the existence of a tree on n vertices whose independence polynomial is symmetric and unimodal as well as the existence of a symmetric and unimodal independence polynomial of degree n of a tree.

Load-bearing premise

The definitions of symmetry and unimodality for independence polynomials of trees are compatible with the structural constraints of trees for some or all n.

Editorial extensions

If this is right

  • When such a tree exists for a given n, the counts of independent sets of complementary sizes must match exactly.
  • Unimodality in the polynomial implies the largest number of independent sets occurs near the middle size.
  • Realization of a symmetric unimodal polynomial of degree n as a tree polynomial shows that the set of tree independence polynomials intersects the set of all such sequences.
  • For n where existence holds, the tree can be constructed so its independent-set distribution satisfies both properties simultaneously.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The existence results could be used to generate families of trees whose independence polynomials avoid certain irregularities seen in general graphs.
  • Further checks on small n by direct computation of all trees would test the boundary cases where existence begins or fails.
  • The same symmetry-unimodality question might be posed for other acyclic graphs such as forests or caterpillars to see if the tree case is special.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 3 minor

Summary. The manuscript studies two existence questions for each integer n ≥ 1: whether there exists a tree on exactly n vertices whose independence polynomial is symmetric (palindromic coefficients) and unimodal, and whether there exists a symmetric and unimodal polynomial of degree n that arises as the independence polynomial of some tree (not necessarily on n vertices). The work proceeds via explicit constructions for small n, analysis of selected tree families, and non-existence arguments where applicable.

Significance. If the existence claims hold via the constructions and arguments, the paper contributes concrete examples and partial characterizations to the study of independence polynomials of trees, clarifying which coefficient sequences are realizable under the structural constraints of trees. The focus on both fixed-order trees and fixed-degree polynomials distinguishes the two questions and may guide further work on unimodal generating functions in graph theory.

minor comments (3)
  1. The abstract states the problems studied but does not summarize the main existence results or the values of n for which affirmative or negative answers are obtained; adding one sentence on the scope of the theorems would improve readability.
  2. In the definitions section, the precise statement of unimodality (strict or weak, and handling of plateaus) should be stated explicitly with reference to the coefficient sequence of I(T,x), as minor variations in definition can affect the constructions.
  3. Figure 1 (or the table of small-n examples) would benefit from an additional column listing the actual independence polynomial for each tree shown, to allow direct verification of symmetry and unimodality.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the supportive summary of our work on symmetric and unimodal independence polynomials of trees, the positive assessment of its significance, and the recommendation for minor revision. No specific major comments were listed in the report, so we have no point-by-point rebuttals to provide. We will incorporate any minor editorial suggestions in the revised version.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: existence study with standard definitions

full rationale

The paper investigates existence of trees on n vertices with symmetric and unimodal independence polynomials (and symmetric unimodal polynomials of degree n arising from trees). It employs the standard definitions of symmetry (palindromic coefficient sequence) and unimodality without any derivation that reduces these properties to the paper's own constructions or fitted inputs. No equations, predictions, or self-citations are load-bearing in a way that creates circularity; the work proceeds via explicit constructions and non-existence arguments for small n and families, which are independent of the existence claims themselves. The derivation chain is self-contained against external combinatorial benchmarks.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

Based solely on the abstract; no specific free parameters, ad-hoc axioms, or invented entities are introduced or mentioned.

assumptions (2)
  • standard math Independence polynomials are defined for any graph and have non-negative integer coefficients.
    Standard definition in graph theory.
  • standard math Symmetry and unimodality are well-defined properties of polynomials with real coefficients.
    Basic algebraic properties.

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Cite this review

Pith. "Pith review of Symmetric and unimodal independence polynomials of trees." pith.science (2026). https://pith.science/paper/2604.18824

@misc{pith2026260418824,
  author       = {Pith},
  title        = {Pith review of: Symmetric and unimodal independence polynomials of trees},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2604.18824}},
  note         = {Machine review of arXiv:2604.18824}
}
abstract

Given $n \geq 1$, we study the existence of a tree on $n$ vertices whose independence polynomial is symmetric and unimodal as well as the existence of a symmetric and unimodal independence polynomial of degree $n$ of a tree.

Figures

Figures reproduced from arXiv: 2604.18824 by the authors.

Figure 1
Figure 1. Rooted graphs (A, r),(B, s) and (C, t) from left to right Notice that ((A, r) ∨ (B, s)) ∨ (C, t) ̸= (A, r) ∨ ((B, s) ∨ (C, t)) as shown in [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. ((A, r) ∨ (B, s)) ∨ (C, t) (left) and (A, r) ∨ ((B, s) ∨ (C, t)) (right) polynomials, for our purposes it is convenient to adopt the following equivalent formulation. Set y = x (1 + x) 2 . Definition 1.3. A polynomial h(x) ∈ R[x] is called γ-positive if there exist an integer d ≥ 0 and nonnegative real numbers γ0, γ1, . . . , γ⌊d/2⌋ such that h(x) = ⌊ X d/2⌋ i=0 γix i (1 + x) d−2i . Equivalently, h(x) = (1 + x) d Γh… view at source ↗
Figure 3
Figure 3. The rooted tree (R19, r) A direct computation gives PR19 (x) = 1 + 19x + 153x 2 + 701x 3 + 2058x 4 + 4112x 5 + 5772x 6 + 5772x 7 + 4112x 8 + 2058x 9 + 701x 10 + 153x 11 + 19x 12 + x 13 = (1 + x) 13(1 + 6y + 9y 2 + 4y 3 + y 4 ) [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: The rooted tree (R20, r) A direct computation gives PR20 (x) = 1 + 20x + 171x 2 + 829x 3 + 2548x 4 + 5255x 5 + 7496x 6 + 7496x 7 + 5255x 8 + 2548x 9 + 829x 10 + 171x 11 + 20x 12 + x 13 = (1 + x) 13(1 + 7y + 16y 2 + 14y 3 + 4y 4 ) and PR20−r(x) = 1 + 19x + 155x 2 + 722x…
Figure 5
Figure 5. Figure 5: The graph C(0, 1, 3, 0, 2, 0) = C(2, 3, 0, 3) Lemma 2.3. Let n ∈ [1, 18]. There is a tree T on n vertices such that PT (x) is symmetric and unimodal when n /∈ {2, 4, 5, 7, 10}. When n ∈ {2, 4, 5, 7, 10}, there is no tree with a symmetric independence polynomial. Proof.…

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Reference graph

Works this paper leans on

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