Characterization of Randomly k-Dimensional Graphs

For an ordered set $W=\{w_1,w_2,...,w_k\}$ of vertices and a vertex $v$ in a connected graph $G$, the ordered $k$-vector $r(v|W):=(d(v,w_1),d(v,w_2),.,d(v,w_k))$ is called the (metric) representation of $v$ with respect to $W$, where $d(x,y)$ is the distance between the vertices $x$ and $y$. The set $W$ is called a resolving set for $G$ if distinct vertices of $G$ have distinct representations with respect to $W$. A minimum resolving set for $G$ is a basis of $G$ and its cardinality is the metric dimension of $G$. The resolving number of a connected graph $G$ is the minimum $k$, such that every $k$-set of vertices of $G$ is a resolving set. A connected graph $G$ is called randomly $k$-dimensional if each $k$-set of vertices of $G$ is a basis. In this paper, along with some properties of randomly $k$-dimensional graphs, we prove that a connected graph $G$ with at least two vertices is randomly $k$-dimensional if and only if $G$ is complete graph $K_{k+1}$ or an odd cycle.Comment: 12 pages, 3 figure

Metric dimension of lexicographic product of some known‎ ‎graphs

‎For an ordered set $W=\{w_1,w_2,\ldots,w_k\}$ of vertices and a vertex $v$ in a connected graph $G$, the ordered $k$-vector $r(v|W):=(d(v,w_1),d(v,w_2),\ldots,d(v,w_k))$ is  called  the (metric) representation of $v$ with respect to $W$, where $d(x,y)$ is the distance between the vertices $x$ and $y$. The set $W$ is called  a resolving set for $G$ if distinct vertices of $G$ have distinct representations with respect to $W$. The minimum cardinality of a resolving set for $G$ is its metric dimension. In this paper, we investigate the metric dimension of the lexicographic product  of graphs $G$ and $H$, $G[H]$, for some known graphs

Applications of Particle Swarm Optimization Algorithm to Solving the Economic Load Dispatch of Units in Power Systems with Valve-Point Effects

Reduction of operating costs in power system in order to return the investment costs and more profitability has vital importance in power industry. Economic Load Dispatch (ELD) is one of the most important issues in reducing operating costs. ELD is formulated as a nonlinear optimization problem with continuous variables within the power plants. The main purpose of this problem is optimal planning of power generation in power plants with minimum cost by total units, regarded to equality and inequality constraints including load demand and the range of units' power productivity. In this article, Economic Load Dispatch problem has been modeled by considering the valve-point loading effects with power plants' constraints such as: the balance of production and consumption in system, the forbidden zones, range of production, increasing and decreasing rates, reliability constraints and network security. To solve the problem, Particle Swarm Optimization (PSO) Algorithms has been employed. To evaluate the effectiveness of the proposed method, the problem has been implemented on a power system with 15 generating units and the results have been evaluated.DOI:http://dx.doi.org/10.11591/ijece.v4i6.672

The effectiveness of worry exposure in treating generalized anxiety disorder

Worry exposure (WE) is a core of cognitive-behavioral treatment for generalized anxiety disorder (GAD). The present study was carried out to examine the efficacy of WE in treating patients with GAD. Three patients with GAD were selected using Structured Clinical Interview for DSM-IV (SCID) based on disorders axis I. Subjects were selected using purposeful sampling, and underwent the treatment after gaining treatment needs. Multiple baseline experimental single case study was used as the method of the present study. The treatment program was carried out for 8 weekly sessions, with a follow up period of 3 months later treatment ending. Subjects completed Pennsylvania State Worry Questionnaire (PSWQ) and Cognitive Avoidance Questionnaire (CAQ). Research findings represented that WE decreased the intensity of GAD symptoms: Pathologic worry and cognitive avoidance. WE has suitable efficacy in treating Generalized Anxiety Disorder and it maintains suitable efficacy in treating GAD