Intersection Graph of a Module

Let $V$ be a left $R$-module where $R$ is a (not necessarily commutative) ring with unit. The intersection graph \cG(V) of proper $R$-submodules of $V$ is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper $R$-submodules of $V,$ and there is an edge between two distinct vertices $U$ and $W$ if and only if $U\cap W\neq 0.$ We study these graphs to relate the combinatorial properties of \cG(V) to the algebraic properties of the $R$-module $V.$ We study connectedness, domination, finiteness, coloring, and planarity for \cG (V). For instance, we find the domination number of \cG (V). We also find the chromatic number of \cG(V) in some cases. Furthermore, we study cycles in \cG(V), and complete subgraphs in \cG (V) determining the structure of $V$ for which \cG(V) is planar

Application of the optimal homotopy asymptotic method for solving the Cauchy reaction-diffusion problem

In this paper, the optimal homotopy asymptotic method is applied on the Cauchy reaction-diffusion problems to check the effectiveness and performance of the method. The obtained solutions show that the OHAM is more effective, simpler and easier than other methods. Moreover, this technique does not require any discretization or linearization and therefore it reduces significantly the numerical computations. The results reveal that the method is explicit

Effect of temperature on the development of Eotetranychus hirsti (Tetranychidae) on fig leaves

International audienceThe fig spider mite, Eotetranychus hirsti is one of the major pests of fig trees worldwide. The effect of temperature on the developmental time and the survival rate of E. hirsti feeding on fig leaves was determined at six constant temperatures of 15, 20, 25, 30, 32 and 35°C. The total developmental time of females (from egg to adult emergence) at the above-mentioned temperatures was 41.29, 24.15, 16.95, 12.35, 10.21 and 10.67 days, respectively. The lower, optimal and upper developmental threshold (Tmin, Topt and Tmax, respectively) and thermal constant (K) of the pest were estimated by ordinary linear and Logan 6 nonlinear models. The lower temperature threshold (Tmin) and thermal constant (K) of the immature stages were estimated to be 9.86°C and 239.48 degree-days (DD), respectively. The Topt and Tmax were estimated to be 34.30 and 35.44°C, respectively. As the temperature increased from 15 to 30°C, the survival rate of immature stages increased from 33.33 to 70.59 %, then decreased and reached 54.91 % at 35°C. Temperature-dependent development data, thermal requirements and temperature thresholds can be used to predict the occurrence, number of generations and population dynamics of E. hirsti

Connectivity of 2-distance graphs

For a simple graph $G$, the $2$-distance graph, $D_2(G)$, is a graph with the vertex set $V(G)$ and two vertices are adjacent if and only if their distance is $2$ in the graph $G$. In this paper, we characterize all graphs with connected 2-distance graph. For graphs with diameter 2, we prove that $D_2(G)$ is connected if and only if $G$ has no spanning complete bipartite subgraphs. For graphs with a diameter greater than 2, we define a maximal Fine set and by contracting $G$ on these subsets, we get a new graph $\hat G$ such that $D_2(G)$ is connected if and only if $D_2(\hat G)$ is connected. Especially, $D_2(G)$ is disconnected if and only if $\hat G$ is bipartite.Comment: 7 pages, 6 figure