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‎Let $G=(V,E)$ be a graph‎. ‎A subset $Ssubset V$ is a hop dominating set‎‎if every vertex outside $S$ is at distance two from a vertex of‎‎$S$‎. ‎A hop dominating set $S$ which induces a connected subgraph‎ ‎is called a connected hop dominating set of $G$‎. ‎The‎‎connected hop domination number of $G$‎, ‎$ gamma_{ch}(G)$,‎‎‎ ‎is the minimum cardinality of a connected hop‎‎dominating set of $G$...

Suppose G is an nvertex and medge simple graph with edge set E(G). An integervalued function f: E(G) → Z is called a flow. Tutte was introduced the flow polynomial F(G, λ) as a polynomial in an indeterminate λ with integer coefficients by F(G,λ) In this paper the Flow polynomial of some dendrimers are computed.

Let k ≥ 1 be an integer, and let G be a finite and simple graph with vertex set V (G). A signed total Italian k-dominating function (STIkDF) on a graph G is a functionf : V (G) → {−1, 1, 2} satisfying the conditions that $sum_{xin N(v)}f(x)ge k$ for each vertex v ∈ V (G), where N(v) is the neighborhood of $v$, and each vertex u with f(u)=-1 is adjacent to a vertex v with f(v)=2 or to two vertic...

Let $kgeq 1$ be an integer, and let $G$ be a graph. A {it$k$-rainbow dominating function} (or a {it $k$-RDF}) of $G$ is afunction $f$ from the vertex set $V(G)$ to the family of all subsetsof ${1,2,ldots ,k}$ such that for every $vin V(G)$ with$f(v)=emptyset $, the condition $bigcup_{uinN_{G}(v)}f(u)={1,2,ldots,k}$ is fulfilled, where $N_{G}(v)$ isthe open neighborhood of $v$. The {it weight} o...

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...

a signed graph (marked graph) is an ordered pair $s=(g,sigma)$$(s=(g,mu))$, where $g=(v,e)$ is a graph called the underlyinggraph of $s$ and $sigma:erightarrow{+,-}$$(mu:vrightarrow{+,-})$ is a function. for a graph $g$, $v(g),e(g)$ and $c(g)$ denote its vertex set, edge set and cut-vertexset, respectively. the lict graph $l_{c}(g)$ of a graph $g=(v,e)$is defined as the graph having vertex set ...