MCA Articles - Page 72 of 102

A graph is a set of points, called nodes or vertices, which are interconnected by a set of lines called edges. The study of graphs, or graph theory is an important part of a number of disciplines in the fields of mathematics, engineering and computer science.Graph TheoryDefinition − A graph (denoted as G = (V, E)) consists of a non-empty set of vertices or nodes V and a set of edges E. A vertex a  represents an endpoint of an edge. An edge joins two vertices a, b  and is represented by set of vertices it connects.Example − Let us ... Read More

Distance between Two VerticesIt is number of edges in a shortest path between Vertex U and Vertex V. If there are multiple paths connecting two vertices, then the shortest path is considered as the distance between the two vertices.Notation − d(U, V)There can be any number of paths present from one vertex to other. Among those, you need to choose only the shortest one.ExampleTake a look at the following graph −Here, the distance from vertex 'd' to vertex 'e' or simply 'de' is 1 as there is one edge between them. There are many paths from vertex 'd' to vertex ... Read More

It is the number of vertices adjacent to a vertex V. Notation − deg(V). In a simple graph with n number of vertices, the degree of any vertices is − deg(v) = n – 1 ∀ v ∈ G A vertex can form an edge with all other vertices except by itself. So the degree of a vertex will be up to the number of vertices in the graph minus 1. This 1 is for the self-vertex as it cannot form a loop by itself. If there is a loop at any of the vertices, then it is not a ... Read More

Whether it is possible to traverse a graph from one vertex to another is determined by how a graph is connected. Connectivity is a basic concept in Graph Theory. Connectivity defines whether a graph is connected or disconnected.ConnectivityA graph is said to be connected if there is a path between every pair of vertex. From every vertex to any other vertex, there should be some path to traverse. That is called the connectivity of a graph. A graph with multiple disconnected vertices and edges is said to be disconnected.Cut VertexLet 'G' be a connected graph. A vertex V ∈ G ... Read More

Whether it is possible to traverse a graph from one vertex to another is determined by how a graph is connected. Connectivity is a basic concept in Graph Theory. Connectivity defines whether a graph is connected or disconnected. It has subtopics based on edge and vertex, known as edge connectivity and vertex connectivity. Let us discuss them in detail.ConnectivityA graph is said to be connected if there is a path between every pair of vertex. From every vertex to any other vertex, there should be some path to traverse. That is called the connectivity of a graph. A graph with ... Read More

Connected GraphA graph is connected if any two vertices of the graph are connected by a path.Vertex 1Vertex 2PATHaba baca b c, a cada b c d, a c dbcb a c , b ccdc dDisconnected GraphA graph is disconnected if at least two vertices of the graph are not connected by a path. If a graph G is disconnected, then every maximal connected subgraph of G is called a connected component of the graph G.Vertex 1Vertex 2PATHaba bacNot AvailableadNot AvailablebcNot Availablecdc d

Two functions f: A → B and g: B → C can be composed to give a composition g o f. This is a function from A to C defined by (g o f)(x) = g(f(x))ExampleLet f(x) = x + 2 and g(x) = 2x + 1, find (f o g)(x) and (g o f)(x).Solution(f o g)(x) = f(g(x)) = f(2x + 1) = 2x + 1 + 2 = 2x + 3(g o f)(x) = g (f(x)) = g(x + 2) = 2 (x+2) + 1 = 2x + 5Hence, (f o g)(x) ≠ (g o f)(x)Some Facts about ... Read More

The inverse of a one-to-one corresponding function f: A → B, is the function g: B → A, holding the following property −f(x) = y ⇔ g(y) = xThe function f is called invertible if its inverse function g exists.ExampleA Function f : Z → Z, f(x)=x+5, is invertible since it has the inverse function g : Z → Z, g(x)= x-5.A Function f : Z → Z, f(x)=x2 is not invertiable since this is not one-to-one as (-x)2=x2.

Let 'G−' be a simple graph with some vertices as that of 'G' and an edge {U, V} is present in 'G−', if the edge is not present in G. It means, two vertices are adjacent in 'G−' if the two vertices are not adjacent in G.If the edges that exist in graph I are absent in another graph II, and if both graph I and graph II are combined together to form a complete graph, then graph I and graph II are called complements of each other.ExampleIn the following example, graph-I has two edges 'cd' and 'bd'. Its complement ... Read More

The rules of mathematical logic specify methods of reasoning mathematical statements. Greek philosopher, Aristotle, was the pioneer of logical reasoning. Logical reasoning provides the theoretical base for many areas of mathematics and consequently computer science. It has many practical applications in computer science like design of computing machines, artificial intelligence, definition of data structures for programming languages etc.Major CategoriesMathematical logics can be broadly categorized into three categories.Propositional Logic − Propositional Logic is concerned with statements to which the truth values, "true" and "false", can be assigned. The purpose is to analyse these statements either individually or in a composite manner.Predicate ... Read More