Identify the feasible cone
Derive the linear inequalities that constrain independence-polynomial coefficients for the W_p class.
← All researchMF-PRISM-MATH-2026-11 / interactive guide
THE IDEA, MADE VISIBLE
Watch the coefficient skyline change.
Count independent choices in a simple clique family before meeting the paper’s connected construction.
Candidate resolutionA Weighted Roller-Coaster Theorem for W_p Graphs
Choose without conflict
An independent set contains no pair joined by an edge. In one complete graph, you can choose no vertex or exactly one vertex.
- Groups
- 4
- Independent sets
- 256
- Largest coefficient
- 108
In this teaching graph, I(x) = (1 + 3x)^4; the coefficients are 1, 12, 54, 108, 81.
[xᵏ](1+px)ᵠ = C(q,k) pᵏ
TRY THIS
Increase the clique size without changing the number of groups. Higher-degree coefficients receive stronger multiplicative weights.
What this experiment represents. Exact counts for a disconnected clique-product teaching family. It is not the connected Wₚ construction or a certificate of the weighted Roller-Coaster theorem.
Mathematics · Preprint
A Weighted Roller-Coaster Theorem for W_p Graphs
Sharp coefficient orderings and connected realizations
Candidate resolutionMF-PRISM-MATH-2026-11 / v1.1
Characterizes a closed cone of independence-polynomial coefficients for connected W_p graphs and constructs arbitrary strict orderings in its sharp tail range. The p = 2 specialization addresses the stated 1-well-covered Roller-Coaster conjecture.
- Identifier
- MF-PRISM-MATH-2026-11
- Series
- Mathematics
- Edition
- Version 1.1
- Length
- 14 pages
- Reserved DOI
- 10.5281/zenodo.22728332 (record reserved)
01
Plain-language guide
3 minute explanation
What this paper is really saying.
The paper proposes a sharp description of which coefficient orderings can occur in independence polynomials of connected W_p graphs and constructs graphs realizing every ordering in the permitted tail range.
graph theoryindependence polynomialsW_p graphs
An independence polynomial counts independent vertex sets by size. Its coefficients can rise and fall in many orders, and the Roller-Coaster problem asks which orderings are actually realizable by well-covered graphs.
The paper first describes a cone of coefficient vectors that W_p graphs must occupy. It then uses approximation and clique expansion to build connected examples whose tail coefficients appear in any prescribed strict order allowed by the theorem.
02
How it works
The argument, without the notation.
The paper’s technical details matter, but the basic route can be understood in three moves.
Approximate a target profile
Transfer a coefficient-approximation theorem through clique expansion to approach a desired ordering.
Realize it with connected graphs
Construct connected W_p graphs and prove the cutoff where arbitrary strict tail orderings become attainable.
03
What it adds
The useful takeaways.
A closed coefficient cone is proposed for connected W_p graphs.
Every strict ordering in the stated sharp tail range is constructively realizable.
For p = 2, the result addresses the unrestricted-order 1-well-covered Roller-Coaster conjecture under the manuscript’s stated interpretation.
04
Scope & limits
What this does—and does not—establish.
Current status
Candidate theorem for connected W_p graphs; the p = 2 case addresses the unrestricted-order 1-well-covered conjecture.
This manuscript presents a candidate resolution. Publication is not verification: independent specialist review remains necessary before the result should be treated as established.
The p = 2 consequence depends on the unrestricted-order interpretation of the original conjecture.
The transfer through the approximation theorem and clique expansion is a central review obligation.
Some certificates represent graphs too large to enumerate directly; the certificate logic must therefore be audited.
05
Why it matters
Why anyone should care.
The theorem would sharply delimit the attainable coefficient orderings and settle the unrestricted-order 1-well-covered case under the stated interpretation.
06
Key terms
The vocabulary, decoded.
These are the terms needed to understand the claim. The full paper uses them more precisely.
Independent set
A set of vertices with no edge joining any pair in the set.
Independence polynomial
A polynomial whose coefficient of x^k counts independent sets of size k.
Well-covered graph
A graph in which all maximal independent sets have the same size.
Clique expansion
A construction that replaces vertices by complete subgraphs to transform coefficient behavior.
07
Independent review
Where scrutiny should concentrate.
The strongest review is not a general reaction. It tests the steps most capable of changing the conclusion.
- 01The exact unrestricted-order interpretation of the original 1-well-covered conjecture.
- 02Transfer of the Cutler–Pebody approximation theorem through clique expansion.
- 03Connected realization, sharp cutoff, and compact certificates for graphs too large to materialize.
Publication record
Go from explanation to evidence.
This page is a reading guide, not a substitute for the manuscript. The public record links the explanation to the paper, source package, review materials, and persistent identifier.
- Document
- Preprint
- Review status
- Open for independent review
- Published
- 2026-09-12
- DOI status
- Reserved · 10.5281/zenodo.22728332
- Canonical record
- MF-PRISM-MATH-2026-11
Metriq PRISM Laboratory, A Weighted Roller-Coaster Theorem for W_p Graphs, Metriq PRISM Laboratory Mathematics Research Paper MF-PRISM-MATH-2026-11, Version 1.1, 2026. Corresponding contributor: Daniel H. Jeffery, ORCID 0009-0001-1200-6042. DOI: 10.5281/zenodo.22728332.