Let $A$ and $B$ be two disjoint subsets of the vertex set $V$ of a graph $G$. The set $A$ is said to dominate $B$, denoted by $A rightarrow B$, if for every vertex $u in B$ there exists a vertex $v in A$ such that $uv in E(G)$. For any graph $G$, a partition $pi = {V_1,$ $V_2,$ $ldots,$ $V_p}$ of the vertex set $V$ is an textit{upper domatic partition} if $V_i rightarrow V_j$ or $V_j rightarrow...

It is well known fact that binary relations are generalized mathematical functions. Contrary to functions from domain to range, binary relations may assign to each element of domain two or more elements of range. Some basic operations on functions such as the inverse and composition are applicable to binary relations as well. Depending on the domain or range or both are fuzzy value fuzzy set, i...

One of the generalizations of the Wiener number to weighted graphs is to assign probabilities to edges, meaning that in nonstatic conditions the edge is present only with some probability. The Reliability Wiener number is defined as the sum of reliabilities among pairs of vertices, where the reliability of a pair is the reliability of the most reliable path. Closed expressions are derived for t...

Molecules are the basic and most important players of life. Molecules are composed of atoms, and each molecule has two main specifications or inheritances that are structures, and in biomedical applications, biological function. This dichotomy can be compared to living organisms or humans whom possess genotype and phenotype. Each of the two main inherited elements contains a variety of feat...

means that every matrix a × b with entries in the set {0, 1, . . . , q − 1} always contains a constant i × j submatrix. The negation of (1) is denoted by (a, b) 6→ (i, j)q . The problem of deciding whether (1) holds was posed by Erdös and Rado [7] (more generally and in a different context), which also can be restated in terms of graph theory or as a variant of the celebrated Ramsey problem. Se...

The chromatic number of a graph G, denoted by χ(G), is the minimum number of colors such that G can be colored with these colors in such a way that no two adjacent vertices have the same color. A clique in a graph is a set of mutually adjacent vertices. The maximum size of a clique in a graph G is called the clique number of G. The Turán graph Tn(k) is a complete k-partite graph whose partition...

In this paper, we study both concepts of geodetic dominatingand edge geodetic dominating sets and derive some tight upper bounds onthe edge geodetic and the edge geodetic domination numbers. We also obtainattainable upper bounds on the maximum number of elements in a partitionof a vertex set of a connected graph into geodetic sets, edge geodetic sets,geodetic domin...

Let G = G(n, n) be a 4–cycle free, bipartite graph on 2n vertices with partitions of equal cardinality n. Let c6(G) denote the number of cycles of length 6 in G. We prove that for n ≥ 4, c6(G) ≤ 13 ( n 2 ) (n − rn), where rn = 12 + √ 4n−3 2 , with equality if and only if G is the incidence point-line graph of a projective plane. Section 1: Introduction. Let G = Gn denote a family of simple grap...

in the mathematical analysis, there are some theorems and definitions that established for both real and fuzzy numbers. in this study, we try to prove  bernoulli's inequality in fuzzy real numbers with some of its applications. also, we prove two other theorems in fuzzy real numbers which are proved before, for real numbers.

Positive coupling effect in gas condensate reservoirs is assessed through a pure theoretical approach. A combination of linear stability analysis and long bubble approximation is applied to describe gas condensate coupled flow and relative permeability, thereof. The role of capillary number in gas condensate flow is clearly expressed through closed formula for relative permeability. While the m...