S.SANTHANAM

  Representation [ edit ] A graph is an abstraction of relationships that emerge in nature; hence, it cannot be coupled to a certain representation. The way it is represented depends on the degree of convenience such representation provides for a certain application. The most common representations are the visual, in which, usually, vertices are drawn and connected by edges, and the tabular, in which rows of a table provide information about the relationships between the vertices within the graph. Visual: Graph drawing [ edit ] Main article:  Graph drawing Graphs are usually represented visually by drawing a point or circle for every vertex, and drawing a line between two vertices if they are connected by an edge. If the graph is directed, the direction is indicated by drawing an arrow. If the graph is weighted, the weight is added on the arrow. A graph drawing should not be confused with the graph itself (the abstract, non-visual structure) as there are several ways to struct...

  Uniform and point wise convergence for sequences of functions [ edit ] Main article:  Uniform convergence In addition to sequences of numbers, one may also speak of  sequences of functions   on   � ⊂ � , that is, infinite, ordered families of functions  � � : � → � , denoted  ( � � ) � = 1 ∞ , and their convergence properties. However, in the case of sequences of functions, there are two kinds of convergence, known as  pointwise convergence  and  uniform convergence , that need to be distinguished. Roughly speaking, pointwise convergence of functions  � �  to a limiting function  � : � → � , denoted  � � → � , simply means that given any  � ∈ � ,  � � ( � ) → � ( � )  as  � → ∞ . In contrast, uniform convergence is a stronger type of convergence, in the sense that a uniformly convergent sequence of functions also converges pointwise, but not conversely. Uniform convergence requires members of the...