Graph theory
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The document introduces the concepts of graphs and different types of graphs. It defines what a graph is - a set of vertices and edges, and how they can be represented. It discusses weighted and non-weighted graphs, directed and undirected graphs, self-loops, multigraphs, and provides examples of how graphs are used in real world applications like social networks, maps, computer networks and more.
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Data structure
This presentation introduces graphs and their types. It will discuss graphs, what they are consisting of nodes and edges. It will also cover different types of graphs like bipartite graphs, complete graphs, and cycle graphs. Finally, it will discuss ways that graphs can be represented in computers, including adjacency matrices and adjacency lists.
Graph therory
introduction and representation of graph.
graph is collection of points and vertices.
there are 2 types of the graph 1. directed
2. undirected graph.
representation of graph 2 ways
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Graphs are non-linear data structures that can be used to represent relationships between objects. A graph contains vertices (nodes) connected by edges (links). A graph G is defined as a set of vertices V connected by a set of edges E. For example, a graph G can be defined as G = (V, E) where V = {A,B,C,D,E} and E = {(A,B), (A,C), (A,D), (B,D), (C,D), (B,E), (E,D)}. Graphs can model real-world networks like roads connecting cities. Euler used graphs to solve the Königsberg bridge problem by showing
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1. Graph Data Structure
A graph is a pictorial representation of a set of objects where some pairs of objects are connected by links. The interconnected objects are represented by points termed as vertices, and the links that connect the vertices are called edges.
Formally, a graph is a pair of sets (V, E), where V is the set of vertices and Eis the set of edges, connecting the pairs of vertices. Take a look at the following graph
V = {a, b, c, d, e}
E = {ab, ac, bd, cd, de}
Lecture 16 graph introduction
1) A graph is a collection of vertices and edges, where vertices represent nodes and edges connect pairs of vertices.
2) Graphs can be undirected, where edges have no directionality, or directed (digraphs), where edges have a direction associated with them.
3) Graphs are commonly represented using adjacency matrices, with cells indicating connections between vertex pairs, or adjacency lists, with each vertex listing its connected neighbors.
2.1 graph basic
A graph is composed of vertices and edges. Vertices are connected by edges. A graph can be directed or undirected. In a directed graph, edges have direction but in an undirected graph edges do not have direction. The degree of a vertex is the number of edges connected to it. A path in a graph is a sequence of adjacent vertices and a cycle is a path from a vertex back to itself. Connectedness refers to whether there is a path between all pairs of vertices in an undirected graph. Graphs can be represented using adjacency matrices or adjacency lists.
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Sub Materi: Persamaan Garis Dari Gambar Garis Lurus & Dari Dua Titik
MATEMATIKA
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GRAPH THEORY
This document defines and provides examples of different types of graphs in graph theory, including:
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- Trivial graphs which have only one vertex.
- Simple graphs which have no parallel edges or loops.
- Undirected graphs whose edges are not directed.
- Directed graphs whose edges are directed with arrows.
- Complete graphs where every pair of vertices is connected by one edge.
- Connected graphs where a path exists between every pair of vertices.
- Disconnected graphs where a path does not exist between every pair of vertices.
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Graph therory
introduction and representation of graph.
graph is collection of points and vertices.
there are 2 types of the graph 1. directed
2. undirected graph.
representation of graph 2 ways
Adjacency matrix
Adjacency list
Data structure graphs
Graphs are non-linear data structures that can be used to represent relationships between objects. A graph contains vertices (nodes) connected by edges (links). A graph G is defined as a set of vertices V connected by a set of edges E. For example, a graph G can be defined as G = (V, E) where V = {A,B,C,D,E} and E = {(A,B), (A,C), (A,D), (B,D), (C,D), (B,E), (E,D)}. Graphs can model real-world networks like roads connecting cities. Euler used graphs to solve the Königsberg bridge problem by showing
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1. Graph Data Structure
A graph is a pictorial representation of a set of objects where some pairs of objects are connected by links. The interconnected objects are represented by points termed as vertices, and the links that connect the vertices are called edges.
Formally, a graph is a pair of sets (V, E), where V is the set of vertices and Eis the set of edges, connecting the pairs of vertices. Take a look at the following graph
V = {a, b, c, d, e}
E = {ab, ac, bd, cd, de}
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1) A graph is a collection of vertices and edges, where vertices represent nodes and edges connect pairs of vertices.
2) Graphs can be undirected, where edges have no directionality, or directed (digraphs), where edges have a direction associated with them.
3) Graphs are commonly represented using adjacency matrices, with cells indicating connections between vertex pairs, or adjacency lists, with each vertex listing its connected neighbors.
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A graph is composed of vertices and edges. Vertices are connected by edges. A graph can be directed or undirected. In a directed graph, edges have direction but in an undirected graph edges do not have direction. The degree of a vertex is the number of edges connected to it. A path in a graph is a sequence of adjacent vertices and a cycle is a path from a vertex back to itself. Connectedness refers to whether there is a path between all pairs of vertices in an undirected graph. Graphs can be represented using adjacency matrices or adjacency lists.
PGL (Persamaan Garis Dari Gambar Garis Lurus & Dari Dua Titik) - Pertemuan 1
Pertemuan 1
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Materi: PGL
Sub Materi: Persamaan Garis Dari Gambar Garis Lurus & Dari Dua Titik
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TP 2021/2022
#jhs
#pjj
#sn
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This document defines and provides examples of different types of graphs in graph theory, including:
- Null graphs which have no edges between vertices.
- Trivial graphs which have only one vertex.
- Simple graphs which have no parallel edges or loops.
- Undirected graphs whose edges are not directed.
- Directed graphs whose edges are directed with arrows.
- Complete graphs where every pair of vertices is connected by one edge.
- Connected graphs where a path exists between every pair of vertices.
- Disconnected graphs where a path does not exist between every pair of vertices.
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This section provides an introduction to graphs and graph theory. Key points include:
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- Common terminology is introduced, such as adjacent vertices, neighborhoods, degrees of vertices, and handshaking theorem.
- Different types of graphs are discussed, including multigraphs, pseudographs, and directed graphs.
- Examples of graph models are given for computer networks, social networks, information networks, transportation networks, and software design. Graphs can be used to model many real-world systems and applications.
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This document discusses graphs and their applications and representations. It defines what a graph is consisting of vertices and edges. It describes different types of graphs such as trees, connected graphs, weighted graphs, directed graphs, and multigraphs. It also discusses graph representations including adjacency matrices and adjacency lists. Finally, it notes that adjacency lists are generally preferred for sparse graphs while adjacency matrices are better for dense graphs.
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This document outlines several postulates of geometry:
1) Postulates are assumed to be true without proof and serve to explain undefined terms and prove other statements.
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3) Five additional postulates are listed relating to line segments, circles, angles, and the intersection of lines cut by a transversal.
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1. Points, lines, and planes are basic geometric objects. A point has no size or dimensions. A line extends indefinitely in both directions and has one dimension. A plane extends indefinitely and has two dimensions.
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3. A line segment connects two points and contains all points between them. A ray starts at an endpoint and extends indefinitely in one direction. Planes and geometric objects are named using points, lines, and planes.
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This document defines and provides examples of key geometric concepts including points, lines, line segments, rays, planes, and their relationships. It defines points as having no dimensions, lines as straight paths extending indefinitely, line segments as parts of lines between two endpoints, and rays as parts of lines extending from an endpoint in one direction. Planes are defined as flat surfaces extending indefinitely. Examples are provided to demonstrate collinear points that lie on the same line, coplanar points that lie in the same plane, and basic postulates such as two points defining a single unique line. Key terms are summarized in a table for easy reference.
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2) It shows that for cyclic polygons with an odd number of sides, the sine rule relating side lengths and opposite interior angles takes the same form as Lami's Theorem.
3) The paper also describes how to transform n-gons into vector diagrams and vice versa, and derives a condition for when a vector diagram can represent a cyclic polygon.
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This document defines and describes different types of graphs. It begins by defining what a graph is as consisting of vertices and edges. It then discusses finite vs infinite graphs, isolated and pendant vertices, null graphs, regular graphs, subgraphs, complete graphs, bipartite graphs, and complete bipartite graphs. Diagrams are provided to illustrate each graph type.
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This document defines key geometric concepts including points, lines, planes and their relationships. It explains that points have no size, lines extend indefinitely and planes are flat surfaces that extend without limits. It also covers topics like collinear points that lie on the same line, determining if objects are coplanar by lying on the same plane, and the different types of intersections between lines and planes, which can be a point, line or no intersection.
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The document discusses shortest path problems and algorithms for finding shortest paths in graphs. It describes Dijkstra's algorithm for finding the shortest path between two nodes in a graph with non-negative edge weights. Prim's algorithm is presented for finding a minimum spanning tree, which is a subgraph connecting all nodes with minimum total edge weight. An example graph is given and steps are outlined for applying Prim's algorithm to find its minimum spanning tree.Skiena algorithm 2007 lecture10 graph data strctures
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Graph theory
- 1. Graph Theory Gaurav Yadav IIIT Sonepat
- 2. Contents ● Introduction to Graphs ● Representation of Graphs ● Weighted Graphs ● Non-weighted Graphs ● Directed Graphs ● Undirected Graphs ● Self-loops ● Multigraphs ● Real World Examples
- 3. Introduction to Graphs ● A graph G, consists of Vertices(V) and Edges(E). ● G = (V,E). ● V is the set of Vertices. ● E is the set of Edges.
- 4. Representation of Graphs ● Let V = {a, b, c}, and E = { { a, b }, { a, c } }. a b c Set describing all the vertices(or nodes) a, b, and c. Set describing an edge between node a and c. |V| = 3 |E| = 2
- 5. Weighted Graphs ● Weight or cost is a numerical value associated to every edge of a graph. We encounter with it in real world, when we need to calculate shortest path between two points, for example – when we see maps to find the shortest driving distance. The path is chosen which has the minimum cost. A E C D B 3 1 1 2 4 2 Shortest path between A & E : A -> D -> C -> E 8 4
- 6. Non-weighted Graphs ● In non weighted graphs, when we doesn’t have weights, all edges are considered equal. The path which has less number of nodes is considered effective. A E C D B Shortest path between A & E : A -> B -> E
- 7. Undirected Graphs ● Undirected graphs have edges that do not have a direction. The edges indicate a two- way relationship, in that each edge can be traversed in both directions. A B D C
- 8. Directed Graphs ● Directed graphs have edges with direction. The edges indicate a one-way relationship, in that each edge can only be traversed in a single direction. A B D C
- 9. Self-loops ● Graphs created can have one or more self-loops, which are edges connecting a node to itself. A B D C
- 10. Multigraphs ● Graphs can have multiple edges with the same source and target nodes, and the graph is then known as a multigraph. A multigraph may or may not contain self-loops. A B D C
- 11. Real World Examples ● Social Graphs – Connections on LinkedIn. ● Path Optimization Algorithms – Google Maps. ● Routing Algorithms in Computer Networks – Routing IP Table in Router. ● Scientific Computations - Edge Chasing in Operating Systems.
- 12. Thank You!