Root geometry of domination polynomials for friendship and book graphs

Abstract: This study examines the domination polynomials of friendship graphs and book graphs, focusing on unanswered questions related to these families. 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 $-2$.

• The paper establishes that domination polynomials for friendship graphs exhibit exactly three real roots (including 0) for even n, with monotonic convergence properties.
• It derives sharp analytic bounds on the modulus of nonzero domination roots and characterizes the limit set of complex roots in book graphs.
• The study resolves integer-root conjectures and introduces a robust analytic framework for asymptotic analysis of graph invariants in dense, symmetric graph families.

Root Geometry of Domination Polynomials for Friendship and Book Graphs

Introduction

The paper "Root geometry of domination polynomials for friendship and book graphs" (2604.08998) addresses open questions concerning the algebraic and asymptotic behavior of domination polynomials for two prominent families of graphs—friendship graphs ($F_n$) and book graphs ($B_n$). The main focus is the fine-grained analysis of the roots of their domination polynomials, specifically the structure, location, and limit behavior of real and complex roots in these graph families. The paper systematically resolves conjectures from prior work and strengthens the theoretical understanding of domination polynomials in dense and locally repetitive graph classes.

Domination Polynomials and Graph Families

A domination polynomial $D(G,x)$ of a graph $G=(V,E)$ enumerates the dominating sets via the generating function

$D(G,x) = \sum_{i=\gamma(G)}^{|V|} d(G,i)x^i,$

where $d(G,i)$ counts dominating sets of size $i$, and $\gamma(G)$ is the domination number. The geometric properties of the roots of $D(G,x)$ (the so-called "domination roots") encode subtle combinatorial and asymptotic features of the underlying graphs.

Friendship graphs ($F_n$) are constructed as $B_n$0, that is, by joining $B_n$1 triangles at a single vertex. Book graphs ($B_n$2) are built by identifying a common edge among $B_n$3 copies of $B_n$4. Both families exhibit high triangle and quadrilateral density, respectively, and have been extensively utilized as benchmarks for algebraic graph invariants.

The known explicit domination polynomials are:

• $B_n$5
• $B_n$6

Real Root Structure of Friendship Graphs

The exact enumeration and location of real roots for $B_n$7, particularly when $B_n$8 is even, was previously unresolved. This paper decisively establishes that for even $B_n$9:

• $D(G,x)$0 has exactly three real roots: $D(G,x)$1 and precisely two simple roots, $D(G,x)$2 in $D(G,x)$3 and $D(G,x)$4 in $D(G,x)$5.
• These roots exhibit monotonic convergence as $D(G,x)$6:

$D(G,x)$7

These results are proved through transformations to logarithmic equations and a detailed monotonicity analysis, rather than relying on numerical arguments. This closes a foundational question about the real-root count for these polynomials.

Upper Bounds on Modulus of Domination Roots for Friendship Graphs

The paper derives a new analytic inequality for any nonzero domination root $D(G,x)$8 of $D(G,x)$9 with $G=(V,E)$0:

$G=(V,E)$1

From this, the explicit modulus bound follows:

$G=(V,E)$2

This sublinear (in $G=(V,E)$3) upper bound is considerably sharper than those achievable via standard coefficient bounds. Empirical evidence shows the actual largest root modulus is consistently less than this bound, but the estimate is rigorous and quantitatively meaningful for the scale of the roots in these families.

Limit Sets and Real Roots in Book Graphs

For book graphs, a complete description of the set of asymptotic root locations (limit set) is obtained via the Beraha–Kahane–Weiss theorem. The limit set $G=(V,E)$4 of domination roots as $G=(V,E)$5 is a union of algebraic curves and isolated points in the complex plane, characterized by:

• Special points: $G=(V,E)$6, $G=(V,E)$7
• Arc of the hyperbola: $G=(V,E)$8
• Arc of the circle: $G=(V,E)$9
• Arc of the quartic: $D(G,x) = \sum_{i=\gamma(G)}^{|V|} d(G,i)x^i,$0

On the real axis, the points of intersection include $D(G,x) = \sum_{i=\gamma(G)}^{|V|} d(G,i)x^i,$1. The parity of $D(G,x) = \sum_{i=\gamma(G)}^{|V|} d(G,i)x^i,$2 (even or odd) determines the exact number and location of real roots, with numerical evidence supporting the following:

• For even $D(G,x) = \sum_{i=\gamma(G)}^{|V|} d(G,i)x^i,$3: four real roots (counting multiplicity), including $D(G,x) = \sum_{i=\gamma(G)}^{|V|} d(G,i)x^i,$4 (multiplicity 2), one root in $D(G,x) = \sum_{i=\gamma(G)}^{|V|} d(G,i)x^i,$5, and one in $D(G,x) = \sum_{i=\gamma(G)}^{|V|} d(G,i)x^i,$6.
• For odd $D(G,x) = \sum_{i=\gamma(G)}^{|V|} d(G,i)x^i,$7: four real roots (counting multiplicity), with two distinct roots in $D(G,x) = \sum_{i=\gamma(G)}^{|V|} d(G,i)x^i,$8 and $D(G,x) = \sum_{i=\gamma(G)}^{|V|} d(G,i)x^i,$9 (multiplicity 2).

Integer Domination Roots

A partial resolution of the integer-root problem is provided:

• Friendship graphs and book graphs have no nonzero integer domination roots.
• For corona-type graph families discussed in earlier work, $d(G,i)$0 is the only nonzero integer domination root.

Explicit computational verification for small $d(G,i)$1 confirms these results and demonstrates that, among all currently tractable families, only the corona constructions realize nonzero integer roots ($d(G,i)$2).

Implications and Future Directions

The paper significantly advances structural understanding of domination polynomials in dense graph families:

• Provides definitive descriptive and quantitative results for real and complex roots of $d(G,i)$3 and $d(G,i)$4, transforming empirically driven conjectures into precise theorems.
• Establishes analytic machinery suitable for root localization in dense families based on explicit polynomial formulas and reduction to algebraic curves in the complex plane.
• The convergence and modulus analysis suggest further avenues for connecting domination-root geometry to invariant-based graph classifications.

Strong numerical evidence points to sharp parity effects and broad regularity in root topologies, but some open problems persist:

• A fully general classification of real root multiplicities for $d(G,i)$5 (especially, resolving Conjecture 3.3) remains to be conclusively established.
• The global integer-root problem — whether $d(G,i)$6 is the only possible nonzero integer root for domination polynomials of arbitrary graphs — continues to attract interest.
• Further refinement of modulus estimates could yield sharper asymptotics and potentially connect to extremal combinatorial invariants.

Methodologically, the analytic inequalities, transformations, and asymptotic root-limit techniques introduced are broadly applicable and likely to prove valuable for other graph polynomial studies, including those for independence and matching polynomials.

Conclusion

This paper conclusively resolves several central questions concerning the root geometry of domination polynomials for friendship and book graph families, including the exact structure of real roots, explicit modulus bounds for complex roots, precise limit set descriptions, and a partial resolution of the integer-root question. The combination of algebraic, analytic, and numerical tools employed constitutes a robust framework for further investigations into algebraic graph invariants and their root behaviors, particularly in dense, symmetric graph families. Open problems regarding global root multiplicities, optimal modulus estimates, and the universal integer-root problem constitute promising directions for continued research in algebraic graph theory (2604.08998).