Some relations between rank of a graph and its complement

Let G be a graph of order n and rank(G) denotes the rank of its adjacency matrix. Clearly, . In this paper we characterize all graphs G such that or n + 2. Also for every integer n ? 5 and any k, 0 ? k ? n, we construct a graph G of order n, such that .     

Commuting decompositions of complete graphs

We say that two graphs G and H with the same vertex set commute if their adjacency matrices commute. In this article, we show that for any natural number r, the complete multigraph K is decomposable into commuting perfect matchings if and only if n is a 2‐power. Also, it is shown that the complete graph K_n is decomposable into commuting Hamilton cycles if and only if n is a prime number. © 2006 Wiley Periodicals, Inc. J Combin Designs     

Synthesis of 14-aryl or alkyl- 14H-dibenzo[a,j]xanthenes promoted by Mg(HSO4)2

14-Aryl or alkyl-14H-dibenzo[a,j]xanthenes were prepared by condensation of 2-naphthol and aldehydes in the presence of Mg（HSO₄）₂ as an efficient,cheap,readily available and eco-friendly catalyst.This procedure has the advantages of high yields,easy work-up,short reaction times,and green conditions.     

A Note on the Roman Bondage Number of Planar Graphs

A Roman dominating function on a graph G = (V(G), E(G)) is a labelling ${f : V(G)\rightarrow \{0,1,2\}}$ satisfying the condition that every vertex with label 0 has at least a neighbour with label 2. The Roman domination number γ _R (G) of G is the minimum of ${\sum_{v \in V(G)}{f(v)}}$ over all such functions. The Roman bondage number b _R (G) of G is the minimum cardinality of all sets ${E\subseteq E(G)}$ for which γ _R (G \ E) > γ _R (G). Recently, it was proved that for every planar graph P, b _R (P) ≤ Δ(P) + 6, where Δ(P) is the maximum degree of P. We show that the Roman bondage number of every planar graph does not exceed 15 and construct infinitely many planar graphs with Roman bondage number equal to 7.     

A rewarding-punishing coordination mechanism based on Trust in a divergent supply chain

Coordination of decentralized supply chains using contract design is a problem that has been widely addressed in the literature. We consider a divergent supply chain including a supplier and several retailers producing fashion products with short sale seasons. The retailers cooperate with the supplier as sales agents; i.e., they work in the framework of revenue sharing contracts. Because of their proximity to the market, retailers can provide more accurate demand forecasts to the supplier that is used to decide on issues such as capacity building and market prices with regard to retailers stiff due dates, different lead times and different price-dependent demand functions. To ensure abundant supply and cope with the demand variability, the retailers have an incentive to exaggerate their private forecast information. In this study, we propose a new rewarding-punishing coordination mechanism based on trust between supply chain tiers, considered as a differentiation factor between honest and deceptive partners. An optimization model is developed as a building block of this mechanism. An approximation method is used to simplify and solve the problem. The model is then implemented using Monte-Carlo simulation in four different situations, according to 10 different strategies for forecast information sharing. The findings from the tests show that the mechanism including trust as a decisional factor performs better than ‘No Trust’ mechanism in all situations. These results suggest that taking into account Trust in designing coordination mechanism may have significant influence on the financial performance of the supply chain.     

A Note on Comaximal Graph of Non-commutative Rings

Let R be a ring with unity. The graph Γ(R) is a graph with vertices as elements of R, where two distinct vertices a and b are adjacent if and only if Ra?+?Rb?=?R. Let Γ₂(R) be the subgraph of Γ(R) induced by the non-unit elements of R. Let R be a commutative ring with unity and let J(R) denote the Jacobson radical of R. If R is not a local ring, then it was proved that:

1. If $\Gamma_2(R)\backslash J(R)$ is a complete n-partite graph, then n?=?2.
2. If there exists a vertex of $\Gamma_2(R)\backslash J(R)$ which is adjacent to every vertex, then R????₂×F, where F is a field.

In this note we generalize the above results to non-commutative rings and characterize all non-local ring R (not necessarily commutative) whose $\Gamma_2(R)\backslash J(R)$ is a complete n-partite graph.     

Some fuzzy stochastic orderings for fuzzy random variables

A new approach to stochastic ordering of fuzzy random variables is investigated in this paper. The traditional definitions of stochastic ordering, hazard rate ordering, and also mean residual life ordering were extended and proposed the unified indexes to ranking fuzzy random variables. Finally, we study the stochastic ordering of fuzzy order statistics by using our proposed approach and established some properties.