Metadata
bibkey: rather2026bookroots
authors: Bilal Ahmad Rather
year: 2026
title: On roots of domination polynomials for friendship and book graphs
doi: null
url: https://arxiv.org/abs/2604.08998v2
claim: "The literal arXiv v1/v2 Conjecture 5.4 classifies all real roots with multiplicities for every shared-edge four-cycle book of order n >= 1; the order-one book contradicts its odd clause."
strata_touched:
- D5/S3/Combinatorics/Graph/Domination/RatherBookRootRefutation
license: citation-only
triage: anchor
Real-root configurations for book domination polynomials
Verified locator
Source: https://arxiv.org/abs/2604.08998v2
The versioned full texts are https://arxiv.org/pdf/2604.08998v1 and https://arxiv.org/pdf/2604.08998v2. The v1 title is Root geometry of domination polynomials for friendship and book graphs; v2 uses the title in the metadata above. Section and printed-page locators below refer to these arXiv versions. The v2 HTML conjecture is at https://arxiv.org/html/2604.08998v2#S5.Thmtheorem4.
The subject here is the location and multiplicity of finitely many real roots, viewed as a configuration on the Euclidean real line. The underlying graph model and its polynomial remain in the graph-domination module.
The graph and ordinary count
Section 2, printed page 3 and Figure 1(b), defines the book B_n for every integer n >= 1 by identifying one common edge in n copies of C4. All other vertices are distinct. Printed page 11 explicitly identifies B_1 with C4. The local n >= 2 hypothesis in Proposition 5.3 and numerical tables do not supply a global n >= 2 hypothesis to Conjecture 5.4.
A subset S dominates a finite simple undirected graph G when every vertex belongs to S or has a neighbor in S. The ordinary domination polynomial is
The formal vertex type is Fin (2n+2). Vertices 0 and 1 are shared, and page i has new vertices 2i+2 and 2i+3, with cycle 0–1–(2i+2)–(2*i+3)–0. Adjacency is precisely these edges and their reversals, without loops. The polynomial sums over the existing DominatingSetAverage.domSets and uses its mem_domSets theorem and the existing SimpleGraph.IsDominating predicate. This is the ordinary count, with no independence or certified-domination restriction.
The entire conjecture
Section 5, Conjecture 5.4, printed page 12:
The numerical data suggest the following sharper parity-dependent statement. (i) If n is even, then D(B_n,x) has exactly four real roots counting multiplicity: 0 with multiplicity 2, one simple root in (-infinity,-2), and one simple root in (-1/2,0). (ii) If n is odd, then D(B_n,x) has exactly four real roots counting multiplicity: 0 with multiplicity 2 and two simple roots in (-1/2,0).
The closed formal claim quantifies over every natural n >= 1 and contains both parity implications. Each branch specifies distinct nonzero real numbers a,b, actual Polynomial.IsRoot predicates, actual rootMultiplicity values 2,1,1, and for every real r the equivalence between being a root and being one of 0,a,b. Thus intervals, exact multiplicities and exhaustiveness are all part of the claim.
The order-one obstruction
For B_1=C4, all six pairs, four triples and the full vertex set dominate. The empty set and four singletons fail. Consequently
For a nonzero real a, the factor a^2 is strictly positive and (a+2)^2+2 is positive. There is therefore no nonzero real root. Applying the odd branch at n=1 would supply such a root and is contradictory. The formal result is the negation of the complete universal claim, using the actual graph equality and finite dominating-set enumeration. The separately amended claim restricted to n >= 2 remains unsettled by this argument. The familiar C4 count and positive quadratic are prior mathematics, not independent new root-classification results.
Source qualification and prior ingredients
The author bibliography at https://sites.google.com/view/bilalahmadrr/publications lists the exact paper as Filomat (2026), to appear. The arXiv record states acceptance by Filomat. The actual journal text, any corrections and any independently revised author manuscript remain unread. The target is the literal arXiv v1/v2 conjecture; agreement with a journal text, absence of a correction and author intent concerning the boundary are not established.
The established book polynomial occurs in arXiv:1401.2092, Theorem 8; Question 3 asks about its roots. ArXiv:1501.05856, Theorem 5, repeats the polynomial and Theorem 10 describes a root-limit set. These are prior ingredients, not the same finite-order parity classification. Jahari and Alikhani, Some families of graphs with no nonzero real domination roots, EJGTA 6 (2018), 17–28, DOI 10.5614/ejgta.2018.6.1.2, treats other graph constructions. Certified domination uses a different invariant.
The relevant six-page article by Saeid Alikhani, Some new results on domination roots of a graph, Electronic Notes in Discrete Mathematics 43 (2013), 425–430, DOI 10.1016/j.endm.2013.07.062, remains unread. Its located arXiv:1210.3144v1 PDF contains only a one-page abstract, and the institutional IPM M/12991 record links an empty TeX document. Neither establishes whether the journal article settles an equivalent complete claim.
The inspected primary sources did not identify an equivalent settlement; this is a bounded literature conclusion, with the named full-text gaps. It supplies neither global novelty nor an exhaustive ownership claim.