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नेटवर्क की अद्भुत दुनिया -16 प्रकार के ग्राफ़ बच्चों को समझाए गए

दिसम्बर 29, 2020

Amazing World of Networks - Graphs Explained To Kids

नेटवर्क क्या होता है?

यह आलेख “नेटवर्क की अद्भुत दुनिया -16 प्रकार के ग्राफ़ बच्चों को समझाए गए”, ग्राफ़ और नेटवर्क में उनके उपयोग के बारे में है। कंप्यूटर शब्दावली में, एक नेटवर्क में दो या अधिक कंप्यूटर होते हैं जो संसाधनों को साझा करने के लिए जुड़े होते हैं (जैसे प्रिंटर और सीडी), फाइलों का आदान-प्रदान, या इलेक्ट्रॉनिक संचार की अनुमति देते हैं।

किसी नेटवर्क के कंप्यूटरों को केबल, टेलीफोन लाइनों, रेडियो तरंगों, उपग्रहों या अवरक्त प्रकाश किरणों (इंफ़्रा रेड लाइट) के माध्यम से जोड़ा जा सकता है। न केवल कंप्यूटर, यहां तक ​​कि अन्य संचार उपकरण जैसे सेल फोन नेटवर्क के माध्यम से जुड़ने के कारण कार्य करते हैं।

एक नेटवर्क कनेक्टेड ऑब्जेक्ट का एक संग्रह होता है। हम वस्तुओं को नोड्स के रूप में उल्लेख करते हैं और आमतौर पर उन्हें बिंदु के रूप में दर्शाते हैं। हम नोड्स के बीच के कनेक्शन का उल्लेख एज (edge) से करते हैं और उसे बिंदुओं के बीच की रेखाओं के रूप में दर्शाते हैं।

नेटवर्क की अद्भुत दुनिया -16 प्रकार के ग्राफ़ बच्चों को समझाए गए
नेटवर्क – वर्टिसिस, एजजेज़

ग्राफ़ क्या होता है?

इकाइयों के एक समूह के बीच परस्पर संबंध दिखाने के लिए नेटवर्क आरेख को ग्राफ़ के रूप में भी जाना जाता है। प्रत्येक इकाई को एक नोड (या शीर्ष) द्वारा दर्शाया जाता है। नोड्स के बीच का संबंध लिंक (या एजजेज़) के माध्यम से दर्शाया गया है।

ग्राफ़ गणितीय संरचनाएं हैं जिनका उपयोग वस्तुओं और इकाइयों के बीच युग्मक संबंधों का अध्ययन करने के लिए किया जाता है। यह असतत गणित की एक शाखा है और कंप्यूटर विज्ञान, रसायन विज्ञान, भाषा विज्ञान, संचालन अनुसंधान (ऑपरेशन्स रिसर्च), समाजशास्त्र, आदि में कई अनुप्रयोगों में इसका उपयोग होता है।

औपचारिक रूप से एक ग्राफ़ G = (V, E) में शीर्षों (vertices) V = {V1, V2, V3, …} और एजजेज़ का सेट E = {E1, E2, E3, …} होता है। The set of unordered pairs of distinct vertices whose elements are called edges of the graph G such that each edge is identified with an unordered pair (Vi, Vj) of vertices.

Terminology Associated with Graph

  • The vertices u and v are called the end vertices of the edge (u, v).
  • If two edges have the same end vertices they are called parallel edges.
  • An edge of the form (u, v) is a loop.
  • A graph is simple if it has no parallel edges and loops.
  • A graph is said to be empty if it has no edges.
  • A graph is a null graph if it has no vertices.
  • A graph with only 1 vertex is a trivial graph.
  • Edges are adjacent if they have a common vertex.
  • Vertices are adjacent if they have a common edge.
  • The degree of the vertex v, written as d(v), is the number of edges with v as an end vertex. By convention, we count a loop twice and parallel edges contribute separately.

Application of Graph

  • Computer Science: In computer science, the graph is used to represent networks of communication, data organization, computational devices, etc.
  • Physics and Chemistry: Graph theory is also used to study molecules in chemistry and physics.
  • Social Science: Graph theory is also widely used in sociology.
  • Mathematics: In this, graphs are useful in geometry and certain parts of topology such as knot theory.
  • Biology: Graph theory is useful in biology and conservation efforts.

Types of Graphs

Graphs are of different types and mainly can be classified as

Null Graph: A null graph is a graph in which there are no edges between its vertices. A null graph is also called an empty graph.

नेटवर्क की अद्भुत दुनिया -16 प्रकार के ग्राफ़ बच्चों को समझाए गए
Null Graph

In the above graph, there are four vertices but not a single edge. It’s an example of a null graph.

Trivial Graph:A trivial graph is a graph that has only one vertex.

नेटवर्क की अद्भुत दुनिया -16 प्रकार के ग्राफ़ बच्चों को समझाए गए
Trivial Graph

In the above graph, there is only one vertex denoted by 1.

Simple Graph: A simple graph is an undirected graph with no parallel edges and no loops. In a simple graph that has n vertices, the degree of every vertex is at most (n – 1). In the above example, the first graph is not a simple graph because it has two edges between A and B and it also has a loop. The second graph is a simple graph because it does not contain any loop and parallel edges.

नेटवर्क की अद्भुत दुनिया -16 प्रकार के ग्राफ़ बच्चों को समझाए गए
Simple Graph
नेटवर्क की अद्भुत दुनिया -16 प्रकार के ग्राफ़ बच्चों को समझाए गए
Not Simple Graph with multiple edges
नेटवर्क की अद्भुत दुनिया -16 प्रकार के ग्राफ़ बच्चों को समझाए गए
Not Simple Graph with loops

Undirected Graph: An undirected graph is a graph whose edges are not directed.

नेटवर्क की अद्भुत दुनिया -16 प्रकार के ग्राफ़ बच्चों को समझाए गए
Undirected Graph

Directed Graph: A directed graph is a graph in which the edges are directed by arrows. Directed graphs are also known as digraphs.

नेटवर्क की अद्भुत दुनिया -16 प्रकार के ग्राफ़ बच्चों को समझाए गए
Directed Graph

In the above graph, each edge is directed by an arrow. A directed edge has an arrow from A to B, meaning A is related to B, but B is not related to A. (In terms of network, there is a path from A to B, but there is no path from B to A.

Complete Graph: A graph in which every pair of vertices is joined by exactly one edge is called a complete graph. It contains all possible edges. In the above example, since each vertex in the graph is connected with all the remaining vertices through exactly one edge therefore, both graphs are complete graphs.

नेटवर्क की अद्भुत दुनिया -16 प्रकार के ग्राफ़ बच्चों को समझाए गए
Complete Graph

Connected Graph: A connected graph is a graph in which we can visit any vertex from any other vertex. In a connected graph, at least one edge or path exists between every pair of vertices.

नेटवर्क की अद्भुत दुनिया -16 प्रकार के ग्राफ़ बच्चों को समझाए गए
Connected Graph

Disconnected Graph: A disconnected graph is a graph in which any path does not exist between every pair of vertices.

नेटवर्क की अद्भुत दुनिया -16 प्रकार के ग्राफ़ बच्चों को समझाए गए
Disconnected Graph

In the above graph there is no path between 3 and 0, 3 and 1, 3 and 2 and also between 4 and 0, 4 and 1 and 4 and 2.

Regular Graph: A regular graph is a graph in which the degree of all the vertices is the same. If the degree of all the vertices is k, then it is called a k-regular graph.

नेटवर्क की अद्भुत दुनिया -16 प्रकार के ग्राफ़ बच्चों को समझाए गए
Regular Graph

In the above graph degree of each vertex is 3. Hence, it’s a 3-regular graph.

Cyclic Graph: A graph with n vertices (where n >= 3) and n edges forming a cycle of n with all its edges is known as a cycle graph. And a graph containing at least one cycle in it is called a cyclic graph.

नेटवर्क की अद्भुत दुनिया -16 प्रकार के ग्राफ़ बच्चों को समझाए गए
Cyclic Graph

In the above graph, vertices B, C, E and D forms a cycle, therefore, it’s a cyclic graph.

Acyclic Graph: A graph that does not have even one cycle is called an acyclic graph.

नेटवर्क की अद्भुत दुनिया -16 प्रकार के ग्राफ़ बच्चों को समझाए गए
Acyclic Graph

Bipartite Graph: A bipartite graph is a graph in which the vertex set can be partitioned into two sets that edges only go between sets, not within them. A graph G(V, E) is called a bipartite graph if its vertex set V(G) can be decomposed into two non-empty disjoint subsets V1(G) and V2(G) in such a way that each edge has its one last joint in V1(G) and the other last point in V2(G).

नेटवर्क की अद्भुत दुनिया -16 प्रकार के ग्राफ़ बच्चों को समझाए गए
Bipartite graph

Star Graph: A star graph is a graph that exactly looks like a star where (n – 1) vertices are connected to a single central vertex. A star graph with n vertices is denoted by Sn.

नेटवर्क की अद्भुत दुनिया -16 प्रकार के ग्राफ़ बच्चों को समझाए गए
Star Graphs

All the above graphs are star graphs, as one vertex is connected to all the remaining vertices.

Weighted Graph: A weighted graph is a graph whose edges have been labeled with some weights or numbers. The length of a path in a weighted graph is the sum of the weights of all the edges in the path.

नेटवर्क की अद्भुत दुनिया -16 प्रकार के ग्राफ़ बच्चों को समझाए गए
Weighted Graph

The above graph is a weighted graph as each edge has been labeled with a value called weight. A graph showing the cities with distances is another example of weighted graph.

Planar Graph: A planar graph is a graph that we can draw in a plane in such a way that no two edges of it cross each other except at a vertex to which they are incident.

नेटवर्क की अद्भुत दुनिया -16 प्रकार के ग्राफ़ बच्चों को समझाए गए
Planar Graph

All the above graphs are planar graphs as none of the edges in all the graphs are crossing each other.

Non-Planar Graph: A graph that cannot be drawn without at least one pair of its crossing edges is known as a non-planar graph. In other words, a graph that is not a planar graph is called a non-planar graph.

नेटवर्क की अद्भुत दुनिया -16 प्रकार के ग्राफ़ बच्चों को समझाए गए
Non Planar Graph

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