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Let G = {g1,...,gn} be a finite abelian group. Consider the complete graph with the vertex set {g1.....,.....g n}. The G-coloring of Kn is a proper edge coloring in which the color of edge {gi,gj} gi g i + gj, 1 ≤ i < 3 ≤ n. We prove that in the G-coloring of the complete graph Kn, there exists a multicolored Hamilton path if G is not an elementary abelian 2-group. Furthermore, we show that if n is odd, then the G-coloring of Kn can be decomposed into multicolored 2-factors and there are exactly lr/2 multicolored r-uniform 2-factors in this decomposition where lr is the number of elements of order r in G, 3 ≤ r ≤ n. This provides a generalization of a recent result due to Constantine which... 

For every h ∈ ℕ, a graph G with the vertex set V (G) and the edge set E(G) is said to be h-magic if there exists a labeling l: E(G) → ℤh{0} such that the induced vertex labeling s: V (G) → ℤh, defined by s(v) = Puv∈E(G) l(uv) is a constant map. When this constant is zero, we say that G admits a zero-sum h-magic labeling. The null set of a graph G, denoted by N(G), is the set of all natural numbers h ∈ ℕ such that G admits a zero-sum h-magic labeling. In 2012, the null sets of 3-regular graphs were determined. In this paper we show that if G is an r-regular graph, then for even r (r > 2), N(G) = ℕ and for odd r (r ≠ 5), ℕ {2, 4} ⊆ N(G). Moreover, we prove that if r is odd and G is a 2-edge... 

Let G be a graph and H be an abelian group. For every subset SH a map φ:E(G)→S is called an S-flow. For a given S-flow of G, and every v∈V(G), define s(v)=∑uv∈E(G)φ(uv). Let k∈H. We say that a graph G admits a k-sum S-flow if there is an S-flow such that for each vertex v,s(v)=k. We prove that if G is a connected bipartite graph with two parts X={x1,⋯,xr}, Y={y1,⋯, ys} and c1,⋯,cr,d1,⋯, ds are real numbers, then there is an R-flow such that s( xi)=ci and s(yj)=dj, for 1≤i≤r,1≤j≤s if and only if ∑i=1rci=∑j=1s dj. Also, it is shown that if G is a connected non-bipartite graph and c1,⋯,cn are arbitrary integers, then there is a Z-flow such that s(vi)=ci, for i=1, ⋯,n if and only if the number... 

Let G be a graph. Assume that l and k are two natural numbers. An l-sum flow on a graph G is an assignment of non-zero real numbers to the edges of G such that for every vertex v of G the sum of values of all edges incident with v equals l. An l-sum k-flow is an l-sum flow with values from the set {±1,…, ±(k — 1)}. Recently, it was proved that for every r, r ≥ 3, r ≠ 5, every r-regular graph admits a 0-sum 5-flow. In this paper we settle a conjecture by showing that every 5-regular graph admits a 0-sum 5-flow. Moreover, we prove that every r-regular graph of even order admits a 1-sum 5-flow  

The energy of a graph G, denoted by E (G), is defined as the sum of the absolute values of all eigenvalues of G. Let G be a graph of order n and rank (G) be the rank of the adjacency matrix of G. In this paper we characterize all graphs with E (G) = rank (G). Among other results we show that apart from a few families of graphs, E (G) ≥ 2 max (χ (G), n - χ (over(G, -))), where n is the number of vertices of G, over(G, -) and χ (G) are the complement and the chromatic number of G, respectively. Moreover some new lower bounds for E (G) in terms of rank (G) are given. © 2008 Elsevier B.V. All rights reserved  

A graph G of order n is called hyperenergetic if E(G) > 2n -2, where E(G) is the energy of G. In this paper it is shown that Kneser graph K n:r is hyperenergetic for any naturals n and r > 2 with n > 2r + 1. Also we prove that for r > 2, the complement of Kneser graph, E(K n.r), is hyperenergetic  

Let G be a graph. The minimum number of colors needed to color the edges of G is called the chromatic index of G and is denoted by x0(G). It is well known that δ(G) ≤ x0(G) ≤ δ(G) + 1, for any graph G, whereδ(G) denotes the maximum degree of G. A graph G is said to be class 1 if x0(G) = δ(G) and class 2 if x0(G) = δ(G)+1. Also, Gδ is the induced subgraph on all vertices of degreeδ(G). Let f : V(G) ! N be a function. An f -coloring of a graph G is a coloring of the edges of E(G) such that each color appears at each vertex v 2 V(G) at most f (v) times. The minimum number of colors needed to f -color G is called the f -chromatic index of G and is denoted by x0f (G). It was shown that for every... 

Suppose that G is a graph and f: V (G) → ℕ is a labeling of the vertices of G. Let S(v) denote the sum of labels over all neighbors of the vertex v in G. A labeling f of G is called lucky if S(u) ≠ S(v) for every pair of adjacent vertices u and v. Also, for each vertex v ∈ V(G), let L(v) denote a list of natural numbers available at v. A list lucky labeling, is a lucky labeling f such that f(v) ∈ L(v) for each v ∈ V(G). A graph G is said to be lucky k-choosable if every k-list assignment of natural numbers to the vertices of G permits a list lucky labeling of G. The lucky choice number of G, ηl(G), is the minimum natural number k such that G is lucky k-choosable. In this paper, we prove that... 

Let G = (V,E) be a simple undirected graph. For a given set L ⊂ ℝ, a function ω: E → L is called an L-flow. Given a vector γ ∈ ℝv, ω is a γ-L-flow if for each υ ∈ V, the sum of the values on the edges incident to υ is γ(υ). If γ(υ) = c, for all υ ∈ V, then the γ-L-flow is called a c-sum L-flow. In this paper, the existence of γ-L-flows for various choices of sets L of real numbers is studied, with an emphasis on 1-sum flows. Let L be a subset of real numbers containing 0 and denote L*:= L {0}. Answering a question from [S. Akbari, M. Kano, and S. Zare. A generalization of 0-sum flows in graphs. Linear Algebra Appl., 438:3629-3634, 2013.], the bipartite graphs which admit a 1-sum ℝ*-flow or... 

Optimal Reactive Power Dispatch (ORPD) has a salient impact on decreasing the power loss of transmission lines and adjusting the voltage deviation. The parameters that are used for the ORPD are tap settings of transformers, voltages of generating plants and the output of compensating device such as capacitor banks and synchronous condensors. In this paper settings of FACTS devices are considered as additional parameters in solving the ORPD problem. Also the genetic algorithm has been used to find the optimal settings of the controlling parameters. The results of the ORPD have been obtained on a 30 bus IEEE network. © 2009 IEEE  

Relativistic electrons, generated in the interaction of an ultra-intense laser pulse with plasma in front of a high-Z solid target, when passing near the nuclei of the solid target produce several MeV highly collimated Bremsstrahlung gamma beam, which can be used to induce photo-nuclear reactions. In this work the possibility of photo-induced transmutation (γn) of a nuclear waste of 126Sn with a half-life of 100,000 years into 125Sn with a half-life of 9.64 days was investigated for the first time. Calculations based on the available experimental data show that the Bremsstrahlung γ beam generated by irradiating a 2 mm thick tantalum target as a converter with 1020Wcm-2μm2 and 10 Hz table-top... 

Let G be a graph. The core of G, denoted by GΔ, is the subgraph of G induced by the vertices of degree Δ(G), where Δ(G) is the maximum degree of G. A k-edge coloring of a graph G is a function f:E(G)L, where |L|=k and f( e1)≠f( e2), for every two adjacent edges e1, e2 of G. The edge chromatic number of G, denoted by χ′(G), is the minimum number k for which G has a k-edge coloring. A graph G is said to be Class 1 if χ′(G)= Δ(G) and Class 2 if χ′(G)=Δ(G)+1. In this paper, it is shown that, for every connected graph of even order, if GΔ= C6, then G is Class 1. Also, we prove that, if G is a connected graph, and every connected component of GΔ is a unicyclic graph or a tree, and GΔ is not a... 

The optimum convolution of dual short pulse for producing the maximum wakefield and the highest dissociation probability of CH4 has been investigated. By using three fundamental shapes of pulses though four different arrangements, the generated wake are considered in plasma. It is found that when the first and second pulses were rectangular-triangular and sinusoidal pulse shapes, respectively, the resultant wakefield amplitude is the highest. This effect opens up a new novel way by pulse shaping mechanism in the photo dissociation dynamics of molecules and controlling of chemical reactions in the desired channels by short pulse intense lasers for reducing the computation time of genetic... 

Let G be a graph. A zero-sum flow of G is an assignment of non-zero real numbers to the edges of G such that the sum of the values of all edges incident with each vertex is zero. Let k be a natural number. A zero-sum k-flow is a flow with values from the set {±1,...,±(k - 1)}. It has been conjectured that every r-regular graph, r ≥ 3, admits a zero-sum 5-flow. In this paper we provide an affirmative answer to this conjecture, except for r = 5  

The possibility of photonuclear transmutation of 93Zr, a highly radioactive nuclear waste with a half-life of 1.53 million years, into 92Zr its stable isotope, through a (γ, n) reaction has been analytically evaluated in this paper. By focusing Intensities more than 1020 W/cm2 onto a solid target, high energy electron generation, Bremsstrahlung and photonuclear reactions have been observed. Using the available data, the number of reactions that produced 92Zr, have been analytically calculated. In addition, this work has shown that the laser intensity, irradiation time and repetition rate of laser have strong and direct effects on the yield of 92Zr and the number of reactions. Irradiating a... 

Nowadays, using Blockchain Technology (BCT) is growing faster in each country. It is essential to apply BCT in Supply Chain Network Design (SCND) and is considered by the designer and manager of SC. This research indicates Viable Supply Chain Network Design (VSCND) by applying BCT. A new form of two-stage robust optimization is suggested. Facility locations and activation BCT for VSCND is the first stage of decisions; finally, we determine flow transshipment between components in the next stage. The GAMS-CPLEX is used for solving the model. The results show that running BCT will decrease 0.99% in costs. There is an economic justification for using BCT when demand is high. A fix-and-optimize... 

Introduction: Among several surgical treatments, the use of transplantation of epidermal cultured melanocytes or melanocytes–keratinocytes cell suspension has gained many researchers and dermatologists' attention as a new technique for the treatment of vitiligo. The present study aimed to transplant autologous epidermal melanocytes–keratinocytes cell suspension for the treatment of vitiligo. Methods: In this study, 15 volunteer patients aged between 18 and 45 years old were studied. The autologous melanocytes–keratinocytes cell suspension was then transplanted to the region after dermabrasion. The included patients were evaluated by VisioFace, MPA9, and Skin Scanner-DUB once before and 1, 2,...