GRAPHS

Graph Theory, in discrete mathematics, is the study of the graph.

A graph is defined as a mathematical structure that represents a particular function by connecting a set of points. It is used to create a  pairwise relationship between objects.

The graph is made up of vertices (nodes) that are connected by the edges (lines). The applications of the linear graph are used not only in Maths but also in other fields such as Computer Science, Physics and Chemistry, Linguistics, Biology, etc. In real-life also the best example of graph structure is GPS, where you can track the path or know the direction of the road.

What is Graph

In Mathematics, a graph is a pictorial representation of any data in an organised manner. The graph shows the relationship between variable quantities.

In a graph theory, the graph represents the set of objects, that are related in some sense to each other. The objects are basically mathematical concepts, expressed by vertices or nodes and the relation between the pair of nodes, are expressed by edges.

History

The history of graph theory states it was introduced by the famous Swiss mathematician named Leonhard Euler, to solve many mathematical problems by constructing graphs based on given data or a set of points.

The graphical representation shows different types of data in the form of bar graphs, frequency tables, line graphs, circle graphs, line plots, etc.

Definition

Graph Theory is the study of points and lines. In Mathematics, it is a sub-field that deals with the study of graphs. It is a pictorial representation that represents the Mathematical truth.

Graph theory is the study of relationship between the vertices (nodes) and edges (lines).

Distance-time graphs

If an object moves along a straight line, the distance travelled can be represented by a distance-time graph.

In a distance-time graph, the gradient of the line is equal to the speed of the object. The greater the gradient (and the steeper the line) the faster the object is moving.

Example

Calculate the speed of the object represented by the green line in the graph, from 0 to 4 s.

change in distance = (8 – 0) = 8 m

change in time = (4 – 0) = 4 s

speed=distancetime

speed=8÷4

speed=2 m/s

Distance-time graphs for accelerating objects – Higher

If the speed of an object changes, it will be accelerating or decelerating. This can be shown as a curved line on a distance-time graph.

The table shows what each section of the graph represents:

Section of graph	Gradient	Speed
A	Increasing	Increasing
B	Constant	Constant
C	Decreasing	Decreasing
D	Zero	Stationary (at rest)

If an object is accelerating or decelerating, its speed can be calculated at any particular time by:

• drawing a tangent to the curve at that time
• measuring the gradient of the tangent

As the diagram shows, after drawing the tangent, work out the change in distance (A) and the change in time (B).

gradient=vertical change(A)horizontal change(B)

It should also be noted that an object moving at a constant speed but changing direction continually is also accelerating. Velocity, a vector quantity, changes if either the magnitude or the direction changes. This is important when dealing with circular motion.

Velocity-Time Graphs

To draw velocity-time graphs, we will use the three equations of motion.