A weighting of the edges of a graph is called irregular if the weighted degrees of the vertices are all different. In this note we show that such a weighting is possible from the weight set {1, 2, . . . , 6dnδ e} for all graphs not containing a component with exactly 2 vertices or two isolated vertices.

An assignment of positive integer weights to the edges of a simple graph G is called irregular if the weighted degrees of the vertices are all different. The irregularity strength, s(G), is the maximal edge weight, minimized over all irregular assignments, and is set to infinity if no such assignment is possible. In this paper, we determine the exact value s(T ) for trees T in which every two v...

‎a set $s$ of vertices of a graph $g=(v,e)$ without isolated vertex‎ ‎is a {em total dominating set} if every vertex of $v(g)$ is‎ ‎adjacent to some vertex in $s$‎. ‎the {em total domatic number} of‎ ‎a graph $g$ is the maximum number of total dominating sets into‎ ‎which the vertex set of $g$ can be partitioned‎. ‎we show that the‎ ‎total domatic number of a random $r$-regular graph is almost‎...

—The localization accuracy of the conventional threedimensional (3D) node localization algorithm based on received signal strength (RSS) is restricted by the random signal strength fluctuation caused by the irregular propagation environments. In this paper, we propose an improved localization algorithm based on a differential RSS distance estimation algorithm to minimize the influence of the p...

The transmission of a vertex v (chemical) graph G is the sum distances from to other vertices in G. If any two have different transmissions, then irregular graph. It shown that for odd number n≥7 there exists chemical tree order n. A construction provided which generates new trees. Two additional families graphs are characterized by property irregularity and sufficient condition guarantee prese...

Let $G$ be a simple graph with vertex set $V$. A double Roman dominating function (DRDF) on $G$ is a function $f:Vrightarrow{0,1,2,3}$ satisfying that if $f(v)=0$, then the vertex $v$ must be adjacent to at least two vertices assigned $2$ or one vertex assigned $3$ under $f$, whereas if $f(v)=1$, then the vertex $v$ must be adjacent to at least one vertex assigned $2$ or $3$. The weight of a DR...