Graphs and Graph Models Further Reading: sec. 10.1 of Rosen 1 A - PowerPoint PPT Presentation

Graphs and Graph Models Further Reading: sec. 10.1 of Rosen 1

A graph is a mathematical structure consisting of: • Mathematical objects we call vertices . • Mathematical objects we call edges , each of which the graph associates with a pair of vertices that are called the endpoints of the edge. 2

A graph is a mathematical structure consisting of: • Mathematical objects we call vertices . • Mathematical objects we call edges , each of which the graph associates with a pair of vertices that are called the endpoints of the edge. 3

A graph is a mathematical structure consisting of: • Mathematical objects we call vertices . • Mathematical objects we call edges , each of which the graph associates with two ( not necessarily distinct) vertices that are called the endpoints of the edge. 4

A graph is a mathematical structure consisting of: • Mathematical objects we call vertices . • Mathematical objects we call edges , each of which the graph associates with two ( not necessarily distinct) vertices that are called the endpoints of the edge. We say an edge is directed if one endpoint is designated as the initial endpoint and one as the terminal endpoint. • In an undirected graph , no edge is directed. • In a directed graph or digraph , every edge is directed. 5

A graph is a mathematical structure consisting of: • Mathematical objects we call vertices . • Mathematical objects we call edges , each of which the graph associates with two ( not necessarily distinct) vertices that are called the endpoints of the edge. We say an edge is directed if one endpoint is designated as the initial endpoint and one as the terminal endpoint. • In an undirected graph , no edge is directed. • In a directed graph or digraph , every edge is directed. 6

A graph is a mathematical structure consisting of: • Mathematical objects we call vertices . • Mathematical objects we call edges , each of which the graph associates with two ( not necessarily distinct) vertices that are called the endpoints of the edge. We say an edge is directed if one endpoint is designated as the initial endpoint and one as the terminal endpoint. • In an undirected graph , no edge is directed. • In a directed graph or digraph , every edge is directed. 7

A graph is a mathematical structure consisting of: • Mathematical objects we call vertices . • Mathematical objects we call edges , each of which the graph associates with two ( not necessarily distinct) vertices that are called the endpoints of the edge. We say an edge is directed if one endpoint is designated as the initial endpoint and one as the terminal endpoint. • In an undirected graph , no edge is directed. • In a directed graph or digraph , every edge is directed. Graphs are often represented by drawings such as: Drawing of an undirected graph that has 6 vertices U , V , W , X , Y , Z and has 14 edges a , b , ..., n . Notation: means "Vertices a and b are the endpoints of edge y ". 8

A graph is a mathematical structure consisting of: • Mathematical objects we call vertices . • Mathematical objects we call edges , each of which the graph associates with two ( not necessarily distinct) vertices that are called the endpoints of the edge. We say an edge is directed if one endpoint is designated as the initial endpoint and one as the terminal endpoint. • In an undirected graph , no edge is directed. • In a directed graph or digraph , every edge is directed. Graphs are often represented by drawings such as: Drawing of an undirected graph that has 6 vertices U , V , W , X , Y , Z and has 14 edges a , b , ..., n . Notation: means "Vertices a and b are the endpoints of edge y ". 9

A graph is a mathematical structure consisting of: • Mathematical objects we call vertices . • Mathematical objects we call edges , each of which the graph associates with two ( not necessarily distinct) vertices that are called the endpoints of the edge. We say an edge is directed if one endpoint is designated as the initial endpoint and one as the terminal endpoint. • In an undirected graph , no edge is directed. • In a directed graph or digraph , every edge is directed. Graphs are often represented by drawings such as: Drawing of an undirected graph that has 6 vertices U , V , W , X , Y , Z and has 14 edges a , b , ..., n . Notation: means "Vertices a and b are the endpoints of edge y ". 10

A graph is a mathematical structure consisting of: • Mathematical objects we call vertices . • Mathematical objects we call edges , each of which the graph associates with two ( not necessarily distinct) vertices that are called the endpoints of the edge. We say an edge is directed if one endpoint is designated as the initial endpoint and one as the terminal endpoint. • In an undirected graph , no edge is directed. • In a directed graph or digraph , every edge is directed. Graphs are often represented by drawings such as: Drawing of an undirected graph that has 6 vertices U , V , W , X , Y , Z and has 14 edges a , b , ..., n . Notation: means "Vertices a and b are the endpoints of edge y ". 11

A graph is a mathematical structure consisting of: • Mathematical objects we call vertices . • Mathematical objects we call edges , each of which the graph associates with two ( not necessarily distinct) vertices that are called the endpoints of the edge. We say an edge is directed if one endpoint is designated as the initial endpoint and one as the terminal endpoint. • In an undirected graph , no edge is directed. • In a directed graph or digraph , every edge is directed. Graphs are often represented by drawings such as: Drawing of an undirected graph that has 6 vertices U , V , W , X , Y , Z and has 14 edges a , b , ..., n . Notation: means "Vertices a and b are the endpoints of edge y ". 12

A graph is a mathematical structure consisting of: • Mathematical objects we call vertices . • Mathematical objects we call edges , each of which the graph associates with a pair of vertices that are called the endpoints of the edge. We say an edge is directed if one endpoint is designated as the initial endpoint and one as the terminal endpoint. • In an undirected graph , no edge is directed. • In a directed graph or digraph , every edge is directed. Graphs are often represented by drawings such as: Drawing of an undirected graph that has 6 vertices U , V , W , X , Y , Z and has 14 edges a , b , ..., n . Notation: means "Vertices a and b are the endpoints of edge y ". 13

Drawing of an undirected graph that has 6 vertices U , V , W , X , Y , Z and has 14 edges a , b , ..., n . Notation: means "Vertices a and b are the endpoints of edge y ". 14

Drawing of an undirected graph that has 6 vertices U , V , W , X , Y , Z and has 14 edges a , b , ..., n . Notation: means "Vertices a and b are the endpoints of edge y ". 15

Drawing of an undirected graph that has 6 vertices U , V , W , X , Y , Z and has 14 edges a , b , ..., n . Notation: means "Vertices a and b are the endpoints of edge y ". Here is a drawing of a directed graph or digraph : Drawing of a digraph that has 6 vertices U , V , W , X , Y , Z and has 14 edges a , b , ..., n . Notation: means "Vertex a is the initial endpoint and vertex b is the terminal endpoint of edge y ". 16

Drawing of an undirected graph that has 6 vertices U , V , W , X , Y , Z and has 14 edges a , b , ..., n . Notation: means "Vertices a and b are the endpoints of edge y ". Here is a drawing of a directed graph or digraph : Drawing of a digraph that has 6 vertices U , V , W , X , Y , Z and has 14 edges a , b , ..., n . Notation: means "Vertex a is the initial endpoint and vertex b is the terminal endpoint of edge y ". 17

Drawing of an undirected graph that has 6 vertices U , V , W , X , Y , Z and has 14 edges a , b , ..., n . Notation: means "Vertices a and b are the endpoints of edge y ". Here is a drawing of a directed graph or digraph : Drawing of a digraph that has 6 vertices U , V , W , X , Y , Z and has 14 edges a , b , ..., n . Notation: means "Vertex a is the initial endpoint and vertex b is the terminal endpoint of edge y ". 18

Note : A graph drawing is not a graph: It's a representation of the mathematical object that is the graph itself. • The relationship of a graph drawing to the graph it depicts is akin to the relationship of a subway map to the subway system it depicts, and also akin to the relationship of a photo of a person to the actual person shown by the photo. • It's very common to refer to a photo of a person as a person: For example, someone might show a photo of his son and say "This is my son." • Similarly, it's very common to refer to a graph drawing as a graph: For example, you can point to a graph drawing and say "This is the graph described on page 4." o This is fine provided we are aware that the drawing is not really the graph. • Just as different photos of the same person may look quite different, different drawings of the same graph may look quite different! 19