Let $G$ be a simple graph with $n$ vertices and $\pm 1$-weights on edges. Suppose that for every edge $e$ the sum of edges adjacent to (including itself) is positive. Then weights over at least $-\frac{n^2}{25}$. Also we provide an example weighted described properties $-(1+o(1))\frac{n^2}{8(1 + \sqrt{2})^2}$.
 The previous best known bounds were $-\frac{n^2}{16}$ $-(1+o(1))\frac{n^2}{54}$...

The first general Zagreb index of a graph $G$ is defined as the sum $\alpha$th powers vertex degrees $G$, where $\alpha$ real number such that $\alpha \neq 0$ and 1$. In this note, for > 1$, we present upper bounds involving chromatic clique numbers graph; an integer \geq 2$, lower bound independence graph.

co-channel interference is a major factor in limiting the capacity and link quality in cellular communications. as the co-channel interference is modeled by lognormal distribution, sum of the co-channel interferences of neighboring cells is represented by the sum of lognormal random variables (rvs) which has no closed-form expression. assuming independent, identically distributed (iid) rvs, the...

For a simple graph G with chromatic number x(G), the Nordhaus-Gaddum inequalities give upper and lower bounds for z(G)•(G ¢) and z(G)+ x(GC). Based on a characterization by Fink of the extremal graphs G attaining the lower bounds for the product and sum, we characterize the extremal graphs G for which A(G)B(G c) is minimum, where A and B are each of chromatic number, achromatic number and pseud...

We solve Diophantine equations of the type $ \, a (x^3 + y^3 z^3 ) = (x y z)^3$, where $x,y,z$ are integer variables, and coefficient $a \neq 0$ is rational. show that there infinite families such equations, including those $a$ any ratio cubes or certain rational fractions, have nontrivial solutions. There also do not solution, $1/a 1 - 24/m$ with restrictions on $m$. The can be represented by ...

We examined how variations in color and brightness are used by the visual system in distinguishing textured surfaces that differed in their first- or second-order statistics. Observers viewed a 32 x 32 array containing two types of square elements differing in chromaticity or luminance or both. The spatial distributions of the two kinds of elements were varied within the array until observers c...