11 problems

Let $G$ be a graph, and let $D(G,x)$ denote its domination polynomial. An integer root is an integer number $r$ such that $D(G,r)=0$. Integer-root conjecture. Every integer root of…

Let $G$ be a graph, let $D_t(G,x)$ denote its total domination polynomial, and let $r$ be an integer root of $D_t(G,x)$. Nasrin's conjecture. For every such $G$ and $r$, one has ……

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 $\eta$ denote the growth rate of the king graph. Numerical computation gives … The bounds … are known. Numerical accuracy conjecture. The first ten decimal digits of the numeri…

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 $C_n$ be the cycle graph with $n$ vertices. For graphs $G$ and $H$, write $G\sim H$ if and only if their domination polynomials satisfy … The equivalence class of $C_n$ is deno…

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…

Few's conjecture. The set of integer domination roots of any graph is a subset of

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…