7 problems

Let $B_n$ be the book graph with parameter $n$, and let $D(B_n,x)$ denote its domination polynomial. The numerical data suggest the following parity-dependent statement. Real-root…

Let $T$ be a tree and let $v\in V(T)$ be a leaf. Let $D(T)$ and $D(T\backslash v)$ denote the domination polynomials of $T$ and the tree obtained by deleting $v$, respectively. Ass…

Let $S$ be a spider, meaning a tree with one vertex of degree greater than $2$, whose legs all have length at most $3$. Let $S'$ be obtained from $S$ by deleting a leaf, and let…

Let $G$ be any graph, and let $D(G)$ denote its domination polynomial, whose coefficient sequence is the sequence of numbers of dominating sets of each cardinality. Alikhani–Peng c…

Let $F_{4,n}$ be the generalized friendship graph with parameter $n$, and consider the complex zeros of its domination polynomial $D(F_{4,n},x)$ as $n$ varies. Limiting-hyperbola c…

Let $F_{4,n}$ be the generalized friendship graph with parameter $n$, and let a domination root mean a real zero of its domination polynomial $D(F_{4,n},x)$. Odd-order nonzero-root…

Aalipour-Hafshejani et al.'s conjecture. For all $a,b\in\mathbb{N}$, if $|a-b|\geq 2$, then ${\cal K}(a,b)$ is ${\cal D}$-unique; that is, every graph with the same domination poly…