فهرست نشریات علمی دسترسی آزاد ایران

The domination polynomial of a graph $G$ of order $n$ is $D(G,x)=\sum_{i=1}^{n} d(G,i) x^i$, where $d(G,i)$ is the number of dominating sets of $G$ of size $i$. For the friendship graph $F_n$, which is the join of $K_1$ with $nK_2$, the domination polynomial is known to be $D(F_n,x)=(2x+x^2)^n+x(1+x)^{2n}$. Motivated by proposed open problems in [S. Alikhani, J.I. Brown, S. Jahari, On the domination polynomials of friendship graphs, Filomat 30:1 (2016), 169-178], we prove that for every even positive integer $n$, $D(F_n,x)$ has exactly three real roots: $x=0$, one root in $(-2,-1)$, and one root in $(-1,0)$. Second, we establish an asymptotic upper bound on the modulus of the complex domination roots of $F_n$: for any root $x$ of $D(F_n,x)$ and for sufficiently large $n$, we have $|x| \leq \sqrt{2n/\ln n} + 1$, so that $|x| = O\left(\sqrt{n/\ln n}\right)$. Furthermore, we address the domination roots of the book graph $B_n$, obtained by gluing $n$ copies of $C_4$ along a common edge. We describe the limiting curve of the roots of $D(B_n,x)$ as $n\to\infty$ and provide an asymptotic bound on their moduli. These results provide a deeper understanding of the nature of domination roots for these important families of graphs.