Found 468 Articles for Mathematics

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 More

A graph can be represented using Adjacency Matrix way.Adjacency MatrixAn Adjacency Matrix A[V][V] is a 2D array of size V × V where $V$ is the number of vertices in a undirected graph. If there is an edge between Vx to Vy then the value of A[Vx][Vy]=1 and A[Vy][Vx]=1, otherwise the value will be zero.For a directed graph, if there is an edge between Vx to Vy, then the value of A[Vx][Vy]=1, otherwise the value will be zero.Adjacency Matrix of an Undirected GraphLet us consider the following undirected graph and construct the adjacency matrix −Adjacency matrix of the above undirected ... Read More

PropositionA proposition is a collection of declarative statements that has either a truth value "true” or a truth value "false". A propositional consists of propositional variables and connectives. We denote the propositional variables by capital letters (A, B, etc). The connectives connect the propositional variables.PredicateA predicate is an expression of one or more variables defined on some specific domain. A predicate with variables can be made a proposition by either assigning a value to the variable or by quantifying the variable.The following are some examples of predicates −Let E(x, y) denote "x = y"Let X(a, b, c) denote "a + ... Read More

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 More

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 More

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 More

A matching graph is a subgraph of a graph where there are no edges adjacent to each other. Simply, there should not be any common vertex between any two edges.MatchingLet 'G' = (V, E) be a graph. A subgraph is called a matching M(G), if each vertex of G is incident with at most one edge in M, i.e., deg(V) ≤ 1 ∀ V ∈ Gwhich means in the matching graph M(G), the vertices should have a degree of 1 or 0, where the edges should be incident from the graph G.Notation − M(G)ExampleIn a matching, if deg(V) = 1, ... Read More

A covering graph is a subgraph which contains either all the vertices or all the edges corresponding to some other graph. A subgraph which contains all the vertices is called a line/edge covering. A subgraph which contains all the edges is called a vertex covering.Line CoveringLet G = (V, E) be a graph. A subset C(E) is called a line covering of G if every vertex of G is incident with at least one edge in C, i.e., deg(V) ≥ 1 ∀ V ∈ Gbecause each vertex is connected with another vertex by an edge. Hence it has a minimum ... Read More

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'.

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 More