Root geometry of domination polynomials for friendship and book graphs
This work addresses theoretical questions in graph theory for researchers studying domination polynomials, but it is incremental as it builds on existing knowledge about these graph families.
The study tackled the problem of determining the real and complex zeros of domination polynomials for friendship and book graphs, showing that for even n, the friendship graph has exactly three real zeros with specific intervals and convergence limits, and provided bounds for complex zeros and results on integer roots.
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 $-2$.