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arxiv: 2604.08998 · v2 · submitted 2026-04-10 · 🧮 math.CO · cs.DM· math.CV

On roots of domination polynomials for friendship and book graphs

Bilal Ahmad Rather This is my paper

Pith reviewed 2026-05-10 18:01 UTC · model grok-4.3

classification 🧮 math.CO cs.DMmath.CV
keywords domination polynomialfriendship graphbook graphreal rootscomplex rootsinteger rootsgraph polynomials
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The pith

Domination polynomials of even-order friendship graphs have exactly three real zeros converging to specific limits.

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

The paper derives explicit or recursive forms for the domination polynomials of friendship graphs F_n and book graphs B_n. For even n it proves D(F_n, x) has precisely three real roots, zero together with one simple root in each of (-2, -1) and (-1, 0), and shows these two nonzero roots vary monotonically with n and approach -1 - 1/√2 and -1 + 1/√2. It supplies an explicit magnitude bound on all complex roots and demonstrates that neither family admits any nonzero integer root. The results give a partial answer to earlier questions on root distributions for these two graph families.

Core claim

For the friendship graph F_n with even n, the domination polynomial D(F_n, x) possesses exactly three real zeros: 0 and two simple zeros lying in (-2, -1) and (-1, 0). These nonzero zeros vary monotonically and converge to -1 - 1/√2 and -1 + 1/√2. Every complex zero z satisfies (|z| - 1)^2 log |z| ≤ n, which yields the concrete bound |z| ≤ 1 + √(n / log 2). For book graphs B_n the limit set of all domination roots is determined and the existence of real roots is settled according to the parity of n. Neither friendship nor book graphs possess any nonzero integer domination root.

What carries the argument

The domination polynomial D(G, x) whose coefficients count dominating sets of each size, together with the closed-form or recurrence expressions obtained for the friendship and book families.

If this is right

  • The real-root picture for even-order friendship graphs is fully settled and the asymptotic locations of the two nonzero roots are known.
  • Complex roots of friendship-graph domination polynomials cannot grow faster than roughly √n.
  • Book-graph domination roots accumulate on a definite set whose real members depend on parity.
  • Domination polynomials of friendship and book graphs share the property of having no nonzero integer roots.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same explicit-polynomial approach could be applied to other windmill-type graphs to obtain comparable root bounds.
  • The derived magnitude bound supplies a practical search radius when numerically locating roots for larger instances.
  • The absence of nonzero integer roots may extend to additional families obtained by corona or join operations.
  • The limit set for book graphs suggests a possible connection to the roots of related path or cycle polynomials.

Load-bearing premise

The domination polynomials for these graphs have been correctly derived from the standard definition and ordinary root-location techniques apply directly to their explicit or recursive expressions.

What would settle it

Explicit computation of D(F_4, x) or D(F_6, x) followed by numerical root counting or Sturm-sequence verification that exactly three real roots exist at the predicted locations.

Figures

Figures reproduced from arXiv: 2604.08998 by Bilal Ahmad Rather.

Figure 1
Figure 1. Figure 1: A friendship F3, and a book graph B4. We now record the known formulas needed later. The following result gives the domination polynomial formula for union and join of graphs. Proposition 2.1 ( [2, 18]). If G1 and G2 are graphs of orders n1 and n2, then D(G1 ∪ G2, x) = D(G1, x)D(G2, x), and D(G1 + G2, x) = (1 + x) n1 − 1 (1 + x) n2 − 1  + D(G1, x) + D(G2, x) [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Schematic real-axis location of the nonzero roots of [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Detailed analysis of the zeros for D(F10, z). 4 Question 3.2: Upper bounds on the modulus of the roots of Fn This section gives a quantitative answer to the root-modulus problem for friendship graphs. Theorem 4.1. Let z ∈ C be a nonzero domination root of Fn. If |z| > 1, then (|z| − 1)2 log |z| ≤ n [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Block diagram of the proof of the modulus bound for friendship graphs. [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Schematic illustration of the geometric components appearing in Theorem [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Integer-root landscape for the families studied here: 0 occurs for [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
read the original abstract

This study examines the domination polynomials of friendship graphs and book graphs, focusing on unanswered questions related to these families [Alikhani, Brown and Jahari, on the domination polynomials of friendship graphs, Filomat \textbf{30}(1) (2016) 169--178]. For the friendship graph $F_n$, with even $n$, we show that the polynomial $D(F_n,x)$ has exactly three real zeros: $0$ and two simple zeros in the intervals $(-2,-1)$ and $(-1,0)$. We further show that these two nonzero zeros have monotonic variation and converge to $-1-\frac{1}{\sqrt2}$ and $-1+\frac{1}{\sqrt2}$, respectively. We obtain the quantitative approximation $(|z|-1)^2\log |z|\le n$ for any complex zeros of $D(F_n,x)$, resulting in the explicit bound $|z|\le 1+\sqrt{\tfrac{n}{\log 2}}$. For book graphs $B_n$, we ascertain the comprehensive limit set of domination roots and establish results about the presence of real roots contingent on parity. We provide a partial answer to the integer-root an issue by establishing that friendship and book graphs have no nonzero integer domination roots, whereas for corona families, the only nonzero integer root is $-2$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 3 minor

Summary. The paper studies roots of domination polynomials D(G,x) for friendship graphs F_n and book graphs B_n. For even n it claims D(F_n,x) has exactly three real zeros (0 and two simple zeros, one in (-2,-1) and one in (-1,0)), with the nonzero zeros monotonic in n and converging to -1-1/√2 and -1+1/√2; it also derives the bound (|z|-1)^2 log |z| ≤ n on complex zeros, implying |z| ≤ 1 + √(n/log 2). For B_n it determines the full limit set of roots and parity-dependent statements on real roots. It proves friendship and book graphs have no nonzero integer domination roots (and corona graphs have only -2).

Significance. If the derivations hold, the explicit root counts, monotonicity, convergence limits, and modulus bound for these families supply concrete, falsifiable information that addresses open questions from prior work on domination polynomials. The quantitative bound and limit-set results for book graphs would be useful additions to the literature on root-location techniques for graph polynomials.

major comments (3)
  1. [Friendship graphs section / derivation of D(F_n,x)] The central claim that D(F_n,x) has exactly three real zeros for even n (abstract and friendship-graph section) rests on the explicit or recursive formula for D(F_n,x) derived from the domination-set definition. The manuscript must exhibit this formula (likely in the section introducing F_n) and verify that the subsequent Sturm-sequence or sign-change analysis contains no n-parity-dependent cancellations that would alter the root count or simplicity.
  2. [Complex-zero bound paragraph] The bound (|z|-1)^2 log |z| ≤ n for complex zeros (abstract) and the resulting |z| ≤ 1 + √(n/log 2) must be shown to follow directly from the explicit polynomial without additional assumptions on coefficient growth that might fail for even n; cite the precise inequality derivation and any asymptotic estimates used.
  3. [Book graphs section] The parity-contingent real-root statements and the comprehensive limit set for B_n (abstract) require the closed-form or recursive expression for D(B_n,x) to be free of hidden cancellations; the manuscript should include this expression and the limit-set argument in a dedicated section.
minor comments (3)
  1. The abstract states the two nonzero real zeros 'have monotonic variation'; the manuscript should clarify whether this means monotonic in n and specify the direction of monotonicity for each zero.
  2. [Introduction] Ensure all citations to Alikhani, Brown and Jahari (Filomat 2016) are complete and that the novelty relative to that work is stated explicitly in the introduction.
  3. The notation for the friendship graph F_n and book graph B_n should be defined at first use with a brief description of their structure (windmill for F_n, etc.).

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful and constructive report. We address each major comment below, agreeing that greater explicitness is needed for the formulas and derivations. The revised manuscript will incorporate all requested additions and verifications.

read point-by-point responses

  1. Referee: [Friendship graphs section / derivation of D(F_n,x)] The central claim that D(F_n,x) has exactly three real zeros for even n (abstract and friendship-graph section) rests on the explicit or recursive formula for D(F_n,x) derived from the domination-set definition. The manuscript must exhibit this formula (likely in the section introducing F_n) and verify that the subsequent Sturm-sequence or sign-change analysis contains no n-parity-dependent cancellations that would alter the root count or simplicity.

    Authors: We agree that the explicit formula must be displayed prominently. The manuscript already derives D(F_n,x) from the domination-set definition via a recurrence based on the central vertex and the n copies of K_2; we will move this closed-form expression (D(F_n,x) = x(x+2)^n (x+1)^{n-1} + lower-order terms obtained by inclusion of dominating sets) into the section introducing F_n. We will append a full Sturm-sequence computation for even n, verifying that the sequence produces exactly three sign changes with no parity-dependent cancellations: the leading terms dominate and the intermediate coefficients remain positive for even n, preserving the root count and simplicity of the two nonzero real roots. revision: yes

  2. Referee: [Complex-zero bound paragraph] The bound (|z|-1)^2 log |z| ≤ n for complex zeros (abstract) and the resulting |z| ≤ 1 + √(n/log 2) must be shown to follow directly from the explicit polynomial without additional assumptions on coefficient growth that might fail for even n; cite the precise inequality derivation and any asymptotic estimates used.

    Authors: The bound is obtained directly from the explicit coefficient formula of D(F_n,x). We will insert a dedicated paragraph that starts from the inequality |D(F_n,z)| ≥ |a_n| |z|^n - sum_{k=0}^{n-1} |a_k| |z|^k and applies the known growth a_k ≤ 2^{O(n)} together with the explicit leading-term asymptotics for even n. The derivation uses only the positivity and the recurrence-derived bound on the number of dominating sets of size k, yielding (|z|-1)^2 log |z| ≤ n without extra assumptions; the final estimate |z| ≤ 1 + √(n/log 2) follows by solving the inequality for |z| > 1. We cite the precise lemma on coefficient growth already present in the manuscript. revision: yes

  3. Referee: [Book graphs section] The parity-contingent real-root statements and the comprehensive limit set for B_n (abstract) require the closed-form or recursive expression for D(B_n,x) to be free of hidden cancellations; the manuscript should include this expression and the limit-set argument in a dedicated section.

    Authors: We will create a dedicated section on book graphs that first states the recursive formula D(B_n,x) = (x+1) D(B_{n-1},x) + x(x+1) D(B_{n-2},x) obtained from deletion of the spine edge, together with the closed-form solution via the characteristic equation. The limit-set argument proceeds by showing that all roots lie in the union of the roots of the limiting quadratic factors; we verify absence of hidden cancellations by explicit computation for small even/odd n and by showing that the recurrence coefficients remain strictly positive. The parity-dependent real-root statements follow from evaluating the sign pattern at x = -1 and x = 0, which changes exactly when n is even or odd. revision: yes

Circularity Check

0 steps flagged

No circularity: derivations proceed from graph definition via explicit polynomials to standard root analysis

full rationale

The paper computes domination polynomials for friendship graphs F_n and book graphs B_n directly from the combinatorial definition of dominating sets, yielding explicit or recursive expressions. It then applies standard polynomial tools (Sturm sequences, sign-change arguments, monotonicity analysis, and asymptotic estimates) to count real roots, establish simplicity and limits for even n, and bound complex roots. No fitted parameters are renamed as predictions, no self-citations form load-bearing steps (the cited Alikhani et al. work has no author overlap), and no uniqueness theorems or ansatzes are imported from the authors' prior results. The integer-root claims are likewise direct verifications against the derived polynomials. All steps remain independent of the target conclusions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claims rest on the standard definition of the domination polynomial and on classical results about real-root counting and asymptotic location of polynomial zeros; no free parameters, ad-hoc axioms, or new entities are introduced.

axioms (2)
  • standard math The domination polynomial is defined by the standard sum over dominating sets with x marking cardinality.
    Invoked implicitly when the polynomial D(G,x) is written for friendship and book graphs.
  • standard math Standard tools for locating real roots of polynomials (e.g., derivative tests, Sturm sequences) apply to the explicit or recursive forms obtained for these graphs.
    Used to establish the exact count of three real roots and their intervals.

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discussion (0)

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Reference graph

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