Article Categories
- All Categories
-
Data Structure
-
Networking
-
RDBMS
-
Operating System
-
Java
-
MS Excel
-
iOS
-
HTML
-
CSS
-
Android
-
Python
-
C Programming
-
C++
-
C#
-
MongoDB
-
MySQL
-
Javascript
-
PHP
Computer Engineering Articles
Page 24 of 35
Mininum spanning tree algorithms
A spanning tree with assigned weight less than or equal to the weight of every possible spanning tree of a weighted, connected and undirected graph $G$, it is called minimum spanning tree (MST). The weight of a spanning tree is the sum of all the weights assigned to each edge of the spanning tree. Following are two most popular algorithms to find a minimum spanning tree (MST).Kruskal's AlgorithmKruskal's algorithm is a greedy algorithm that finds a minimum spanning tree for a connected weighted graph. It finds a tree of that graph which includes every vertex and the total weight of ...
Read MoreMathematical Foundation Introduction
Mathematics can be broadly classified into two categories −Continuous Mathematics − It is based upon continuous number line or the real numbers. It is characterized by the fact that between any two numbers, there are almost always an infinite set of numbers. For example, a function in continuous mathematics can be plotted in a smooth curve without breaks.Discrete Mathematics − It involves distinct values; i.e. between any two points, there are a countable number of points. For example, if we have a finite set of objects, the function can be defined as a list of ordered pairs having these objects, ...
Read MoreMathematical Logical Terms and Definitions
TautologiesA Tautology is a formula which is always true for every value of its propositional variables.Example − Prove [ (A → B) ∧ A ] → B is a tautologyThe truth table is as follows −ABA → B(A → B) ∧ A[ (A → B) ∧ A ] → BTrueTrueTrueTrueTrueTrueFalseFalseFalseTrueFalseTrueTrueFalseTrueFalseFalseTrueFalseTrueAs we can see every value of [ (A → B) ∧ A ] → B is "True", it is a tautology.ContradictionsA Contradiction is a formula which is always false for every value of its propositional variables.Example − Prove (A ∨ B) ∧ [ ( ¬ A) ∧ (¬ B) ] ...
Read MoreMathematical Logical Connectives
A Logical Connective is a symbol which is used to connect two or more propositional or predicate logics in such a manner that resultant logic depends only on the input logics and the meaning of the connective used.Generally there are five connectives which are −OR (∨)AND (∧)Negation/ NOT (¬)Implication / if-then (→)If and only if (⇔).OR (∨) − The OR operation of two propositions A and B (written as A ∨ B) is true if at least any of the propositional variable A or B is true.The truth table is as follows −ABA ∨ BTrueTrueTrueTrueFalseTrueFalseTrueTrueFalseFalseFalseAND (∧) − The AND operation ...
Read MoreKirchoff's Theorem
Kirchoff's theorem is useful in finding the number of spanning trees that can be formed from a connected graph.ExampleThe matrix 'A' be filled as, if there is an edge between two vertices, then it should be given as '1', else '0'.
Read MoreIsomorphism and Homeomorphism of graphs
IsomorphismIf two graphs G and H contain the same number of vertices connected in the same way, they are called isomorphic graphs (denoted by G ≅ H).It is easier to check non-isomorphism than isomorphism. If any of these following conditions occurs, then two graphs are non-isomorphic −The number of connected components are differentVertex-set cardinalities are differentEdge-set cardinalities are differentDegree sequences are differentExampleThe following graphs are isomorphic −HomomorphismA homomorphism from a graph G to a graph H is a mapping (May not be a bijective mapping) h: G → H such that − (x, y) ∈ E(G) → (h(x), h(y)) ∈ ...
Read MoreHamiltonian Graphs
Hamiltonian graph - A connected graph G is called Hamiltonian graph if there is a cycle which includes every vertex of G and the cycle is called Hamiltonian cycle. Hamiltonian walk in graph G is a walk that passes through each vertex exactly once.Dirac's Theorem - If G is a simple graph with n vertices, where n ≥ 3 If deg(v) ≥ {n}/{2} for each vertex v, then the graph G is Hamiltonian graph.Ore's Theorem - If G is a simple graph with n vertices, where n ≥ 2 if deg(x) + deg(y) ≥ n for each pair of non-adjacent ...
Read MoreInverse of function of Set
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.
Read MoreHomomorphism
Two graphs G1 and G2 are said to be homomorphic, if each of these graphs can be obtained from the same graph 'G' by dividing some edges of G with more vertices. Take a look at the following example −Divide the edge 'rs' into two edges by adding one vertex.The graphs shown below are homomorphic to the first graph.If G1 is isomorphic to G2, then G is homeomorphic to G2 but the converse need not be true.Any graph with 4 or less vertices is planar.Any graph with 8 or less edges is planar.A complete graph Kn is planar if and ...
Read MoreEulerian Graphs
Euler Graph - A connected graph G is called an Euler graph, if there is a closed trail which includes every edge of the graph G.Euler Path - An Euler path is a path that uses every edge of a graph exactly once. An Euler path starts and ends at different vertices.Euler Circuit - An Euler circuit is a circuit that uses every edge of a graph exactly once. An Euler circuit always starts and ends at the same vertex. A connected graph G is an Euler graph if and only if all vertices of G are of even degree, ...
Read More