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Mathematics Articles
Page 5 of 21
Hamiltonian Graphs
A Hamiltonian graph is a connected graph that contains a cycle which visits every vertex exactly once and returns to the starting vertex. This cycle is called a Hamiltonian cycle. A Hamiltonian path (or walk) passes through each vertex exactly once but does not need to return to the starting vertex. Unlike Eulerian graphs (which require traversing every edge), Hamiltonian graphs focus on visiting every vertex. Sufficient Conditions for Hamiltonian Graphs There is no simple necessary and sufficient condition to determine if a graph is Hamiltonian. However, two important theorems provide sufficient conditions − Dirac's Theorem ...
Read MoreIsomorphism and Homeomorphism of graphs
In graph theory, isomorphism and homomorphism are ways to compare the structure of two graphs. Isomorphism checks whether two graphs are structurally identical, while homomorphism is a more relaxed mapping that preserves adjacency but does not require a one-to-one correspondence. Isomorphism Two graphs G and H are called isomorphic (denoted by G ≅ H) if they contain the same number of vertices connected in the same way. Formally, there must exist a bijective function f: V(G) → V(H) such that two vertices are adjacent in G if and only if their images are adjacent in H. Checking ...
Read MoreCardinality of a Set
The cardinality of a set S, denoted by |S|, is the number of elements in the set. This number is also referred to as the cardinal number. If a set has an infinite number of elements, its cardinality is ∞. Examples of Cardinality |{1, 4, 3, 5}| = 4 (finite set with 4 elements) |{1, 2, 3, 4, 5, ...}| = ∞ (infinite set of natural numbers) |{}| = 0 ...
Read MoreFunctions of Set
A function assigns to each element of a set, exactly one element of a related set. Functions find their application in various fields like representation of the computational complexity of algorithms, counting objects, study of sequences and strings, to name a few. Function − Definition A function or mapping (defined as f: X → Y) is a relationship from elements of one set X to elements of another set Y (X and Y are non-empty sets). X is called the Domain and Y is called the Codomain of function f. Function f is a relation on X ...
Read MoreFinding the number of regions in the graph
In a connected planar graph, the plane is divided into distinct areas called regions (or faces), including the outer unbounded region. The number of regions can be found using Euler's formula for planar graphs, which relates vertices, edges, and regions. Key Formulas Sum of Degrees Theorem − The sum of the degrees of all vertices equals twice the number of edges − ∑ deg(Vi) = 2|E| Euler's Formula − For any connected planar graph − |V| + |R| = |E| + 2 Where |V| is the number of vertices, |E| ...
Read MoreFinding the simple non-isomorphic graphs with n vertices in a graph
Two graphs are isomorphic if one can be transformed into the other by renaming its vertices. In other words, they have the same structure even if the vertices are labeled differently. Non-isomorphic graphs are graphs that have genuinely different structures − no renaming of vertices can make one look like the other. When counting simple non-isomorphic graphs with n vertices, we look for all structurally distinct graphs possible, ignoring vertex labels. Problem Statement How many simple non-isomorphic graphs are possible with 3 vertices? Solution With 3 vertices, there are at most ⌈3C2⌉ = 3 possible ...
Read MoreComposition of Functions of Set
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)) In composition, the output of the first function becomes the input of the second function. The function on the right (f) is applied first, and then the function on the left (g) is applied to the result. A B C ...
Read MoreIntroduction to Mathematical Logic
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 MoreConstructing Triangles, ASA
Introduction Constructing ASA triangles explains to construct a triangle where the measure of two angles and one of the side lengths is given to us. Geometry is the branch of mathematics that deals with properties & relations of points, lines, surfaces & solids. Also, it deals with geometrical construction. Geometrical construction is constructing or drawing geometrical figures like lines, line segments, triangles, circles & quadrilaterals etc. There are several methods of constructing geometrical figures. We can construct geometrical figures by using geometrical instruments like ruler, protractor, compass, divider & set squares. Triangles are mostly constructed by using a ruler, protractor ...
Read MoreContinuity and Discontinuity
Introduction Continuity and discontinuity can be defined as properties of functions that are used in statistics to predict or estimate values. Mathematical functions can be categorised into two types − continuous and discontinuous variables. Continuity is the property of functions that can be shown on a graph without breaking. Discontinuity, on the other hand, is a property of functions which have unconnected points on a graph. In this tutorial, we will understand the concept of continuity and discontinuity along with examples. Continuity In nature, continuity is seen all around us. For example, the flow of rivers, time etc., are some ...
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