On roots of domination polynomials for friendship and book graphs

TL;DR

This paper investigates the roots of domination polynomials for friendship and book graphs, revealing their real zeros, asymptotic behavior, and bounds on complex zeros, thus advancing understanding of their algebraic properties.

Contribution

It provides new insights into the real and complex roots of domination polynomials for friendship and book graphs, including bounds and limit sets, addressing open questions in the field.

Findings

Friendship graphs with even n have exactly three real zeros of their domination polynomial.

The two nonzero zeros of $D(F_n,x)$ vary monotonically and converge to specific limits.

Complex zeros satisfy explicit bounds, with no nonzero integer roots 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 [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 $-2$.