Author: admin
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Isomorphic Graphs
Consider a graph G(V, E) and G* (V*,E*) are said to be isomorphic if there exists one to one correspondence i.e. f:V→V* such that {u, v} is an edge of G if and only if {f(u), f(v)} is an edge of G*. Number of vertices of graph (a) must be equal to graph (b), i.e.,…
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Representation of Graphs
There are two principal ways to represent a graph G with the matrix, i.e., adjacency matrix and incidence matrix representation. (a)Representation of the Undirected Graph: 1. Adjacency Matrix Representation: If an Undirected Graph G consists of n vertices then the adjacency matrix of a graph is an n x n matrix A = [aij] and defined…
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Types of Graphs
Null Graph: A null graph is defined as a graph which consists only the isolated vertices. Example: The graph shown in fig is a null graph, and the vertices are isolated vertices. 2. Undirected Graphs: An Undirected graph G consists of a set of vertices, V and a set of edge E. The edge set contains the unordered…
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Graph
Graph G consists of two things: 1. A set V=V(G) whose elements are called vertices, points or nodes of G. 2. A set E = E(G) of an unordered pair of distinct vertices called edges of G. 3. We denote such a graph by G(V, E) vertices u and v are said to be adjacent…
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Conditional Probability
Theorem: If A and B are two dependent events then the probability of occurrence of A given that B has already occurred and is denoted by P(A/B) is given by Similarly, the probability of occurrence of B given that A has already occurred is given by Proof: Let S be the sample space.Then, we have Interchange A…
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Multiplication Theorem
Theorem: If A and B are two independent events, then the probability that both will occur is equal to the product of their individual probabilities. P(A∩B)=P(A)xP(B) Proof: Let event A can happen is n1ways of which p are successful B can happen is n2ways of…
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Addition Theorem
Theorem1: If A and B are two mutually exclusive events, then P(A ∪B)=P(A)+P(B) Proof: Let the n=total number of exhaustive cases n1= number of cases favorable to A. n2=…
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Probability
The word ‘Probability’ means the chance of occurring of a particular event. It is generally possible to predict the future of an event quantitatively with a certain probability of being correct. The probability is used in such cases where the outcome of the trial is uncertain. Probability Definition: The probability of happening of an event…
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Generating Functions
Generating function is a method to solve the recurrence relations. Let us consider, the sequence a0, a1, a2….ar of real numbers. For some interval of real numbers containing zero values at t is given, the function G(t) is defined by the series G(t)= a0, a1t+a2 t2+⋯+ar tr+…………equation (i) This function G(t) is called…
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Total Solution
The total solution or the general solution of a non-homogeneous linear difference equation with constant coefficients is the sum of the homogeneous solution and a particular solution. If no initial conditions are given, obtain n linear equations in n unknowns and solve them, if possible to get total solutions. If y(h) denotes the homogeneous solution of…