Matematika Diskrit | Graf Bagian II - 01 : Graf Lengkap, Graf Lingkaran, Graf Teratur, Graf Bipartit

Summary

This video lecture on discrete mathematics focuses on various types of graphs in graph theory. Key topics include complete graphs, where every vertex is connected to every other vertex; circular graphs, where each vertex has a degree of 2; and regular graphs, where all vertices have the same degree. The lecture also covers bipartite graphs, which consist of two disjoint sets of vertices, with edges connecting vertices from one set to another. The video explains how to calculate the number of edges in these graphs and provides practical examples to illustrate each concept.

Takeaways

• 😀 Complete graphs (K_n) are simple graphs where each vertex is connected to every other vertex. The number of edges in a complete graph with n vertices is calculated as n(n-1)/2.
• 😀 Circular graphs (C_n) consist of vertices where each vertex has degree 2, forming a circular arrangement with n vertices.
• 😀 Regular graphs are graphs where all vertices have the same degree, denoted as r. The formula for calculating the number of edges in a regular graph is n * r / 2.
• 😀 In a complete graph with n vertices, each vertex has a degree of n-1, and each edge connects two vertices, forming a highly connected structure.
• 😀 A graph is called bipartite if its vertices can be divided into two disjoint sets such that every edge connects a vertex in one set to a vertex in the other set.
• 😀 The degree of a vertex in a complete graph with n vertices is n-1. For instance, K2 has 1 edge, K3 has 3 edges, and K4 has 6 edges.
• 😀 A graph is regular if each vertex has the same degree. For example, if a graph has 8 vertices and each vertex has a degree of 3, it forms a regular graph.
• 😀 The formula for the number of edges in a regular graph is derived by multiplying the number of vertices (n) by the degree of each vertex (r), and then dividing by 2.
• 😀 In bipartite graphs, there are no edges connecting vertices within the same set, only between the two sets, forming a division between them.
• 😀 Graphs can be categorized based on their structure and connectivity. Examples include complete, circular, regular, and bipartite graphs, each with unique properties and applications.

Q & A

What is a complete graph, and how is it denoted?

A complete graph is a simple graph where every node is connected to every other node. It is denoted as Kₙ, where n is the number of nodes in the graph.

How do you calculate the number of edges in a complete graph with n nodes?

The number of edges in a complete graph with n nodes is calculated using the formula: (n(n-1))/2.

What is a cycle graph, and how is it represented?

A cycle graph is a graph where each node has exactly 2 edges, and it forms a closed loop or cycle. It is denoted as Cₙ, where n is the number of nodes in the graph.

What distinguishes a regular graph from other types of graphs?

A regular graph is one in which every node has the same degree, meaning each node is connected to the same number of edges.

How do you calculate the number of edges in a regular graph?

The number of edges in a regular graph with n nodes and degree R for each node is given by the formula: (n × R)/2.

What is the formula for calculating the number of edges in a complete graph with n nodes?

The formula for the number of edges in a complete graph is (n(n-1))/2, where n is the number of nodes.

What is a bipartite graph, and how do you identify one?

A bipartite graph consists of two disjoint sets of nodes, where edges only connect nodes from one set to nodes in the other set. A graph is bipartite if its nodes can be partitioned into two sets where no two nodes within the same set are connected.

What is the key property of a graph to be considered bipartite?

The key property of a bipartite graph is that every edge connects a node from one set to a node in the other set, with no edges connecting nodes within the same set.

Can a regular graph also be a complete graph? Explain.

Yes, a regular graph can also be a complete graph. For instance, a K₄ (complete graph with 4 nodes) is also a 3-regular graph because every node has 3 edges.

How do you calculate the number of vertices in a regular graph with a fixed number of edges and a given degree?

To calculate the number of vertices in a regular graph, use the formula n = (2 × number of edges) / degree of each vertex. The total degree of the graph is twice the number of edges, and each vertex has the same degree.