➠The word graph is used in mathematics in two quite distinct senses.
➠You probably met it for the first time in the context of graphs of functions;
the usual curves that we draw in the plane to represent the behavior of real-valued functions.
➠In the context of combinatorics, graph has a quite different meaning, which is now explained
➠In computer science, graphs are used to represent networks of communication, data organization, computational devices, the flow of computation, etc
One practical example: The link structure of a website could be represented by a directed graph
➠Graph theory is also used to study molecules in chemistry and physics
➠Graph theory is also widely used in sociology as a way, for example, to measure actors' prestige through the use of social network analysis software
➠Likewise, graph theory is useful in biology and conservation efforts where a vertex can represent regions where certain species exist (or habitats) and the edges represent migration paths, or movement between the regions. This information is important when looking at breeding patterns or tracking the spread of disease, parasites or how changes to the movement can affect other species.
⇛In combinatorics a graph consists of a set of points, called vertices (or, by some people, nodes), some of which are joined by lines, called edges. OR
⇛A graph G is a triple consisting of a vertex set V(G), an edge set E(G), and a relation that associates with each edge two vertices (not necessarily distinct) called its endpoints.
Here are some examples of graphs:
➠Points where the lines in our pictures cross are not necessarily vertices of the graph:
➠In the fourth example above, there are only five vertices, and two edges cross at a point where there is no vertex of the graph.
➠If the vertex u is connected to the vertex v, then v is also connected to u, and hence this relation of being connected is a symmetric relation.
For instance, the competition between species in an ecosystem can be modeled using a niche overlap graph.
Each species is represented by a vertex.
An undirected edge connects two vertices if the two species represented by these vertices compete (that is, some of the food resources they use are the same).
A niche overlap graph is a simple graph because no loops or multiple edges are needed in this model.
For example, we can use a simple graph to represent whether two people know each other, that is, whether they are acquainted.
➠You probably met it for the first time in the context of graphs of functions;
the usual curves that we draw in the plane to represent the behavior of real-valued functions.
➠In the context of combinatorics, graph has a quite different meaning, which is now explained
➠In computer science, graphs are used to represent networks of communication, data organization, computational devices, the flow of computation, etc
One practical example: The link structure of a website could be represented by a directed graph
➠Graph theory is also used to study molecules in chemistry and physics
➠Graph theory is also widely used in sociology as a way, for example, to measure actors' prestige through the use of social network analysis software
➠Likewise, graph theory is useful in biology and conservation efforts where a vertex can represent regions where certain species exist (or habitats) and the edges represent migration paths, or movement between the regions. This information is important when looking at breeding patterns or tracking the spread of disease, parasites or how changes to the movement can affect other species.
⇛In combinatorics a graph consists of a set of points, called vertices (or, by some people, nodes), some of which are joined by lines, called edges. OR
⇛A graph G is a triple consisting of a vertex set V(G), an edge set E(G), and a relation that associates with each edge two vertices (not necessarily distinct) called its endpoints.
Here are some examples of graphs:
How to draw graph !!!
➠In general, we visualize graphs by using points to represent vertices and line segments, possibly curved, to represent edges, where the endpoints of a line segment representing an edge are the points representing the endpoints of the edge.➠Points where the lines in our pictures cross are not necessarily vertices of the graph:
➠In the fourth example above, there are only five vertices, and two edges cross at a point where there is no vertex of the graph.
➠If the vertex u is connected to the vertex v, then v is also connected to u, and hence this relation of being connected is a symmetric relation.
MODELS OF GRAPH
Graphs are used in a wide variety of models.Niche Overlap Graphs
in Ecology Graphs are used in many models involving the interaction of different species of animals.For instance, the competition between species in an ecosystem can be modeled using a niche overlap graph.
Each species is represented by a vertex.
An undirected edge connects two vertices if the two species represented by these vertices compete (that is, some of the food resources they use are the same).
A niche overlap graph is a simple graph because no loops or multiple edges are needed in this model.
Acquaintanceship Graphs
We can use graph models to represent various relationships between people.For example, we can use a simple graph to represent whether two people know each other, that is, whether they are acquainted.
Influence Graphs
In studies of group behavior it is observed that certain people can influence the thinking of others. A directed graph called an influence graph can be used to model this behaviour.The Web Graph
The World Wide Web can be modeled as a directed graph where each Web page is represented by a vertex and where an edge starts at the Web page a and ends at the Web page b if there is a link on a pointing to b.
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