The topological ordering algorithm sorts nodes of a directed Graph such that the order of the tail of each arc is lower than the order of its head. In this paper, we introduce the notion of covering between nodes of a directed Graph. Then, we apply the topological ordering algorithm on Graphs containing the covering nodes. We show that there exists a cut set with forward arcs in these Graphs and the order of the covering nodes is successive.

The independence Graph Ind(G) of a Graph G is the Graph with vertices as maximum independent sets of G and two vertices are adjacent, if and only if the corresponding maximum independent sets are disjoint. In this work, we find the independence Graph of Cartesian product of d copies of complete Graphs Kq, which is known as the Hamming Graph H(d, q). Greenwell and Lovasz [7] found that the independence number of direct product of d copies of Kq as qd−1. We prove that the independence number of Hamming Graph H(d, q), which is cartesian product of d copies of Kq, is also qd−1. As an application of our findings, we find answers for rook problem in higher dimensional square chess board.

In this paper, a neural network is trained for optimal nodal ordering of Graphs to obtain a small wavefront using soft computing. A preference function consists of six inputs that can be seen as a generalization of Sloan's function. These six inputs represent the different connection characteristics of Graph models. This research is done with the aim of comparing Sloan's theoretical numbering method with Sloan's developed method with neural networks and WSA meta-heuristic algorithm. Unlike the Sloan algorithm, which uses two fixed coefficients, six coefficients are used here, based on the evaluation of artificial neural networks. The weight of networks is obtained using Water Strider algorithm. Examples are included to demonstrate the performance of the present hybrid method.

Topological aspects of meshless element free Galerkian method (EFGM) as the nodal connectivity of matrices are studied. Four different Graphs are defined for meshless discretizations of EFGM and their connectivity is investigated. Graphs are used for nodal ordering of meshless models in order to reduce the bandwidth, profile, maximum front width, and root-mean-square wave front of the corresponding matrices.      

In this paper, we compare four ordering policies in a lost sales inventory model with zero ordering cost, constant lead time, and Poisson demand process. These ordering policies are 1) base stock policy, 2) full delay policy, 3) simple delay policy and 4) a recently developed ordering policy called (1; T) or one for one period policy. Our work can be considered as an expansion of a previous research which compared the first three policies. We show that, for any fixed value of the ratio of unit lost sales cost over unit holding cost, there is a specific value of lead time demand beyond which the cost of (1; T) policy is lower than the costs of the other three policies. Furthermore, the superiority of (1; T) policy is more significant for low values of the above ratio and becomes more pronounced as the lead time demand increases.

With considering a ‘triangle of three major currency pairs’, there is a tiny difference between multiplication of exchange rate for the first two currency pairs and the third. To discover whether this little difference can lead to a neutral arbitrage or not, I took portfolios of 35 baskets of three major currency pairs (combinations of all 7 major currencies). There are eight approaches (different cases of short and long positions) in each basket; for example buying 1st currency pairs and selling two others, etc.Historical monthly FX rates were gathered from January 1990 until July 2011. Profit or loss derived from eight approaches in all baskets has been calculated. Number of months with a profit has been compared with the months with a loss. Covariances of all approaches of FX rates between growth rates were calculated.I found that this ‘triangle ordering’ of three currency pairs will not always eventuate to a profit. In the other meaning, this is not a neutral strategy. The results showed that in 94% of 280 cases, the probability of gaining profit is almost equal to gaining loss. Also it was found that the most profitable approach is not the best in probability of profit.Standard deviations of results (as indicators of risk of the approaches) were diagramed with the amount of profits. This figured ‘efficient frontier’ of approaches, the best combinations of risks and profits.Covariance were often positive (in 70% of 280 cases), showing probability of simultaneous effectiveness of external factors on all currencies