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Inverse of function of Set
The inverse of a one-to-one (bijective) function f: A → B is the function g: B → A that reverses the mapping of f. It holds the following property − f(x) = y ⇔ g(y) = x The function f is called invertible if its inverse function g exists. For a function to be invertible, it must be one-to-one (injective) − meaning no two different inputs map to the same output − and onto (surjective) − meaning every element in the codomain is mapped to by some element in the domain. The inverse of f is commonly denoted as f−1. ...
Read MoreConnected vs Disconnected Graphs
The article is already well-structured. I'll replace the two JPG images with SVG diagrams and keep everything else intact. In graph theory, graphs are classified as connected or disconnected based on whether there exists a path between every pair of vertices. Understanding this distinction is fundamental to analyzing graph structure and its applications. Connected Graph A graph is connected if there exists a path between any two vertices in the graph. In other words, starting from any vertex, you can reach every other vertex by traversing edges. Connected Graph ...
Read MoreConnectivity of Graph
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 that defines whether a graph is connected or disconnected. It has subtopics based on edge and vertex, known as edge connectivity and vertex connectivity. Connectivity A graph is said to be connected if there is a path between every pair of vertices. A graph with vertices that cannot reach each other is said to be disconnected. Example 1: Connected Graph In the following graph, it is possible ...
Read MoreCut Set and Cut Vertex of Graph
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 that defines whether a graph is connected or disconnected. Connectivity A graph is said to be connected if there is a path between every pair of vertices. A graph with multiple disconnected vertices and edges is said to be disconnected. Cut Vertex Let G be a connected graph. A vertex V ∈ G is called a cut vertex (or articulation point) of G if removing V (and ...
Read MoreDegree of Vertex of a Graph
The degree of a vertex V is the number of edges incident with (connected to) that vertex. Notation − deg(V) In a simple graph with n vertices, the maximum degree of any vertex is − deg(v) ≤ n - 1, for all v in G A vertex can form an edge with all other vertices except itself. So the degree of a vertex will be at most the number of vertices minus 1. If there is a loop at any vertex, then it is not a simple graph. Degree of a vertex ...
Read MoreDistance between Vertices and Eccentricity
The article is already well-structured. I'll replace the JPG image with an SVG diagram and keep everything else intact. In graph theory, the distance between two vertices and the eccentricity of a vertex are fundamental concepts used to measure how far apart vertices are within a graph. These concepts lead to the definitions of the radius and diameter of a graph. Distance between Two Vertices The distance between two vertices U and V is the number of edges in the shortest path between them. If there are multiple paths connecting two vertices, the shortest one is considered ...
Read MoreEdges and Vertices of Graph
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 Definition 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 represents an endpoint of an edge. An edge joins two vertices and is represented by the set of vertices it connects. Example ...
Read MoreFinding the chromatic number of complete graph
The chromatic number of a graph is the minimum number of colors needed to color its vertices such that no two adjacent vertices share the same color. For a complete graph Kn, every vertex is connected to every other vertex, which makes it a special and straightforward case for graph coloring. Problem Statement What is the chromatic number of the complete graph Kn? Solution The following diagram shows a properly colored complete graph K4, where each of the 4 vertices requires a different color − Coloring of K₄ (χ = 4) ...
Read MoreFinding the line covering number of a graph
The line covering number (also called the edge cover number) of a graph is the minimum number of edges required to cover all the vertices of the graph. An edge cover is a set of edges such that every vertex in the graph is an endpoint of at least one edge in the set. The line covering number is denoted by α1. Lower Bound Formula For a graph with n vertices, the line covering number has the following lower bound − α1 ≥ ⌈n / 2⌉ This is because each edge can cover ...
Read MoreFinding the matching number of a graph
A matching in a graph is a set of edges where no two edges share a common vertex. The matching number of a graph is the maximum number of edges in any matching − in other words, the largest set of edges you can select such that no vertex appears more than once. The matching number is denoted by β1. Upper Bound For a graph with n vertices, the matching number has the following upper bound − β1 ≤ ⌊n / 2⌋ This is because each edge in a matching uses exactly 2 vertices, and no vertex can ...
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