Mathematics | Graph Theory Basics - Set 1 Last Updated : 14 Apr, 2025 Comments Improve Suggest changes Like Article Like Report A Graph is just a way to show connections between things. It is set of edges and vertices where each edge is associated with unordered pair of vertices. Graph is a data structure that is defined by two components :Node or Vertex: It is a point or joint between two lines like people, cities, or websites. In below diagram the nodes are A, B, C, D, E, F.Edge: It is line or connection between two nodes like connections between them (friendships, roads, links).In the below diagram edges are the connecting lines in between them.Simple GraphTo know about "Graph representation" click here Basic Concepts in GraphOrdered Pair: Ordered pair is a connection between two nodes u and v which is identified by unique pair (u, v). The pair (u, v) is ordered because (u ,v) is not same as (v, u). It is used in case of directed graph to show which vertex is directing to which vertex.Unordered Pair: In this (u, v) that is identified by unique pair(u, v) can be identified as (v, u). In this the order does not matter in which they come, they are treated same. Undirected graphs are its common example.Weighted Graph: It is a graph (directed or undirected) in which each edge is assigned some numerical value. This value is called a weight. These weights often represent costs, distances, capacities or other quantifiable relationships between vertices.Unweighted Graph: It is a graph in which edges do not have any weight assigned. In this graph all the edges are treated equally or given equal priority. There are only two possibility for edges, either an edge exists or it does not.ApplicationsGraph is a data structure which is used extensively in our real-life like examples below:Social Network: Each user is represented as a node and all their activities, suggestion and friend list are represented as an edge between the nodes.Google Maps: Various locations are represented as vertices or nodes and the roads are represented as edges and graph theory is used to find shortest path between two nodes.Recommendations on e-commerce websites: The “Recommendations for you” section on various e-commerce websites uses graph theory to recommend items of similar type to user’s choice.Graph theory is also used to study molecules in chemistry and physics.Terminologies in GraphsAdjacent Node: A node ‘v’ is said to be adjacent node of node ‘u’ if and only if there exists an edge between ‘u’ and ‘v’.Degree of a Node: In an undirected graph the number of edges incident on a node is the degree of the node. In case of directed graph: Indegree of the node is the number of arriving edges to a node. Outdegree of the node is the number of departing edges to a node. Self-Loop: When an edge in graph connects a vertex to itself it is called self-loop. This edge starts and ends at same vertex. A self-loop is counted twice in case of degree of a node.The sum of degree of all the vertices in a graph G is even.Path: It is a sequence of edges which connects a sequence of distinct vertices. In this sequence of edges no vertices is repeated except in case of closed path, where only first and last vertex can be repeated.Isolated Node: A node with degree 0 is known as isolated node. Isolated node can be found by Breadth first search(BFS). It finds its application in LAN network in finding whether a system is connected or not. Types of GraphsTypes of GraphsDirected Graph A graph in which the direction of the edge is defined to a particular node is a directed graph.Directed GraphDirected Acyclic graph: It is a directed graph with no cycle. For a vertex ‘v’ in DAG there is no directed edge starting and ending with vertex ‘v’. The arrows go in one direction only (Directed) and You can’t go in a circle or loop (Acyclic). Tree: A tree is just a restricted form of graph. That is, it is a DAG with a restriction that a child can have only one parent.Undirected GraphA graph in which the direction of the edge is not defined. So if an edge exists between node ‘u’ and ‘v’, then there is a path from node ‘u’ to ‘v’ and vice-versa.Undirected GraphConnected graph: A graph is connected when there is a path between every pair of vertices. In a connected graph there is no unreachable node.Complete graph: A graph in which each pair of graph vertices is connected by an edge. In other words, every node ‘u’ is adjacent to every other node ‘v’ in graph ‘G’. A complete graph would have n(n-1)/2 edges.Biconnected graph: A connected graph which cannot be broken down into any further pieces by deletion of any vertex. It is a graph with no articulation point. Some Important Graphs1. Regular Graph: A graph in which every vertex x has same/equal degree. K-regular graph means every vertex has k edges. Every complete graph Kn will have (n-1)-regular graph which means degree is n-1. Regular graphs 2. Bipartite Graph: It is graph G in which vertex set can be partitioned into two subsets U and V such that each edge of G has one end in U and another end point in V.Bipartite graph 3. Complete Bipartite graph : It is a simple graph with vertex set partitioned into two subsets u and w. U = {v1, v2 , v3, ..., vm} and W = {w1, w2, w3, ..., wn}The elements in these sets are vertices.There is an edge from each vi to each wj.There is no self loop. Complete Bipartite graph4. Cycle graph : It is a connected graph where each vertex has degree 2, forming a single closed loop without any branches or end points. This graph contain atleast 3 vertices. Suppose a graph has following vertices:v1, v2, v3, ..., vn This graph will be cycle graph if it has edges as follows:(v1,v2), (v2,v3), (v3,v4), ..., (vn-1,vn), (vn,v1).Cycle graph Comment More infoAdvertise with us Next Article What are Numbers? G geeksforgeeks user Follow Improve Article Tags : Mathematics graph-cycle Discrete Mathematics Similar Reads Maths Mathematics, often referred to as "math" for short. It is the study of numbers, quantities, shapes, structures, patterns, and relationships. It is a fundamental subject that explores the logical reasoning and systematic approach to solving problems. Mathematics is used extensively in various fields 5 min read Basic ArithmeticWhat are Numbers?Numbers are symbols we use to count, measure, and describe things. 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A set is simply a collection of distinct elements, such as numbers, letters, or even everyday objects, that share a common property or rule.Example of SetsSome examples of sets include:A set of fruits: {apple, 3 min read PracticeNCERT Solutions for Class 8 to 12The NCERT Solutions are designed to help the students build a strong foundation and gain a better understanding of each and every question they attempt. This article provides updated NCERT Solutions for Classes 8 to 12 in all subjects for the new academic session 2023-24. The solutions are carefully 7 min read RD Sharma Class 8 Solutions for Maths: Chapter Wise PDFRD Sharma Class 8 Math is one of the best Mathematics book. It has thousands of questions on each topics organized for students to practice. RD Sharma Class 8 Solutions covers different types of questions with varying difficulty levels. The solutions provided by GeeksforGeeks help to practice the qu 5 min read RD Sharma Class 9 SolutionsRD Sharma Solutions for class 9 provides vast knowledge about the concepts through the chapter-wise solutions. These solutions help to solve problems of higher difficulty and to ensure students have a good practice of all types of questions that can be framed in the examination. Referring to the sol 10 min read RD Sharma Class 10 SolutionsRD Sharma Class 10 Solutions offer excellent reference material for students, enabling them to develop a firm understanding of the concepts covered. in each chapter of the textbook. As Class 10 mathematics is categorized into various crucial topics such as Algebra, Geometry, and Trigonometry, which 9 min read RD Sharma Class 11 Solutions for MathsRD Sharma Solutions for Class 11 covers different types of questions with varying difficulty levels. Practising these questions with solutions may ensure that students can do a good practice of all types of questions that can be framed in the examination. This ensures that they excel in their final 13 min read RD Sharma Class 12 Solutions for MathsRD Sharma Solutions for class 12 provide solutions to a wide range of questions with a varying difficulty level. With the help of numerous sums and examples, it helps the student to understand and clear the chapter thoroughly. Solving the given questions inside each chapter of RD Sharma will allow t 13 min read Like