‎In this paper‎, ‎we investigate domination number‎, ‎$gamma$‎, ‎as well‎ ‎as signed domination number‎, ‎$gamma_{_S}$‎, ‎of all cubic Cayley‎ ‎graphs of cyclic and quaternion groups‎. ‎In addition‎, ‎we show that‎ ‎the domination and signed domination numbers of cubic graphs depend‎ on each other‎.

For any integer $kgeq 1$ and any graph $G=(V,E)$ with minimum degree at least $k-1$‎, ‎we define a‎ ‎function $f:Vrightarrow {0,1,2}$ as a Roman $k$-tuple dominating‎ ‎function on $G$ if for any vertex $v$ with $f(v)=0$ there exist at least‎ ‎$k$ and for any vertex $v$ with $f(v)neq 0$ at least $k-1$ vertices in its neighborhood with $f(w)=2$‎. ‎The minimum weight of a Roman $k$-tuple dominatin...

Let G be a finite and simple graph with vertex set V (G), and let f : V (G) → {−1, 1} be a two-valued function. If k ≥ 1 is an integer and ∑ x∈N(v) f(x) ≥ k for each v ∈ V (G), where N(v) is the neighborhood of v, then f is a signed total k-dominating function on G. A set {f1, f2, . . . , fd} of distinct signed total k-dominating functions on G with the property that ∑d i=1 fi(x) ≤ j for each x...

A two-valued function f defined on the vertices of a graph G (V, E), I : V -+ {-I, I}, is a signed dominating function if the sum of its function values over any closed neighborhood is at least one. That is, for every v E V, f(N[v]) 2: 1, where N(v] consists of v and every vertex adjacent to v. The of a signed dominating function is ICV) = L f( v), over all vertices v E V. The signed domination...

A signed edge domination function (or SEDF) of a simple graph $G=(V,E)$ is $f: E\rightarrow \{1,-1\}$ such that $\sum_{e'\in N[e]}f(e')\ge 1$ holds for each $e\in E$, where $N[e]$ the set edges in $G$ share at least one endpoint with $e$. Let $\gamma_s'(G)$ denote minimum value $f(G)$ among all SEDFs $f$, $f(G)=\sum_{e\in E}f(e)$.In 2005, Xu conjectured $\gamma_s'(G)\le n-1$, $n$ order $G$. Thi...