A Weighted Roller-Coaster Theorem for W_p Graphs

What is this paper trying to establish?

Which coefficient orderings can independence polynomials of connected W_p graphs realize?

The central idea.

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.

What changes if the method holds?

The theorem would sharply delimit the attainable coefficient orderings and settle the unrestricted-order 1-well-covered case under the stated interpretation.

Where scrutiny should concentrate.

The paper identifies the following points as the highest-value targets for independent review.

1. 01
   The exact unrestricted-order interpretation of the original 1-well-covered conjecture.
2. 02
   Transfer of the Cutler–Pebody approximation theorem through clique expansion.
3. 03
   Connected realization, sharp cutoff, and compact certificates for graphs too large to materialize.