Applications of Graph: Statistics is the branch of mathematics that involves collecting, organising, interpreting, presenting and analysing data. Based on the studies of data obtained, people can draw conclusions, make decisions and plan wisely.

Since pictures are good visual aids and leave a long-lasting effect on the mind of an observer, the information contained in numerical data can be easily understood if we represent it in the form of diagrams or graphs. A picture is better than a thousand words. This quote correctly fits with the graphs. Usually, comparisons among the individuals are best shown through graphs. In this article, we will learn about the application of graphs.

Graphs and Its Types

The pictorial representation of data or information is called a graph. There are various ways of representing numerical data graphically.

1. Bar graph
2. Histograms
3. Pie graph
4. Frequency polygon graph
5. Line graph
6. Linear graph

Applications of Graphs to Solve Real-World Problems

Graphs are widely used in many fields. Let us take some real-life examples and solve them through graphs.

Ten Applications of Graphs

Since graphs are powerful abstractions, they can be essential in modelling data. Given below are some instances for the applications of graphs.

• 1. Social network graphs: Graphs show who knows who, how they communicate with one other, and how they impact each other, as well as other social structure relationships. The Facebook graph showing who follows whom or who sends friend invitations to whom is an example. These can be used to figure out how information spreads, how topics get popular, how communities grow, and even who could be a good fit for that person.
• 2. Graphs in epidemiology: Nowadays, the spread of disease is widespread worldwide. Analysing such graphs has become an essential component in understanding and controlling the spread of diseases.
• 3. Utility graphs: The power grid, the Internet, and the water network are graphs with vertices representing connection points and edges representing the cables or pipes that connect them. Understanding the reliability of such utilities under failure or assault and lowering the costs of building infrastructure that meets desired needs requires analysing the features of these graphs.
• 4. Transportation networks: In transportation network graphs, vertices are intersections in road networks, and edges are the road segments that connect them. Vertices are stops in public transit networks, and edges are the links that connect them. Many map systems, such as Google Maps, Bing Maps, and Apple iOS \(6\) maps, employ such networks to determine the optimal paths between sites. They’re also utilised to figure out how traffic flows and how traffic lights work.
• 5. Constraint graphs: Graphs are often used to represent constraints among items. For example, the GSM network for cell phones consists of a collection of overlapping cells. Any pair of cells that overlap must operate at different frequencies. These constraints can be modelled as a graph where the cells are vertices and edges are placed between overlapping cells.
• 6. Dependence graphs: Graphs can be used to represent dependencies or precedences among items. Such graphs are often used in large projects in laying out what components rely on other components and used to minimise the total time or cost to completion while abiding by the dependences.
• 7. Document link graphs: The most well-known example is the web’s link graph, in which each web page is a vertex, and a directed edge connects each hyperlink. Link graphs are used to assess the relevancy of online pages, the most acceptable sources of information, and good link sites, among other things.
• 8. Finite element meshes: Many models of physical systems in engineering entail partitioning space into discrete pieces, such as the flow of air over a vehicle or plane wing, the spread of earthquakes through the ground, or the structural vibrations of a building. The elements and the connections between them make up a graph known as a finite element mesh.
• 9. Robot planning: The edges represent possible transitions between states, whereas the vertices represent various states for the robot. This requires approximating continuous motion as a sequence of discrete steps. Such graph plans are used to plan paths for autonomous vehicles, for example.
• 10. Graphs in compilers: Graphs are used extensively in compilers. They can be used for type inference, so-called data flow analysis, register allocation, and many other purposes. They are also used in specialised compilers, such as query optimisation in database languages.

Solved Examples on Application of Graphs

Q.1. The bar graph shows the expenditure and revenue of a company for each quarter in its first year of operation.

a) Describe the company performance in the third quarter of the year.
b) Was the company profitable in its first year of operation?
Ans: a) In the third quarter, expenditure \( = \) revenue \(=₹ 30\) crores
b) Total expenditure in the four quarters \( =₹ \left({25 + 35 + 30 + 28} \right)\) crores
\( = ₹118\) crores.
Total revenue in the four quarters \( =₹ \left({10 + 12 + 30 + 35} \right)\) crores
\(= ₹87\) crores.
Total expenditure \( – \)Total revenue \(=₹ 118 -₹ 87 =₹ 31\) crores.
Hence, the company was not profitable in its first year of operation.

Q.2. A man with a monthly salary of \(₹6400\) plans his budget for a month as given below.

Item	Food	Clothing	Education	Miscellaneous	Savings
Amount (in \(₹\))	\(2100\)	\(600\)	\(1200\)	\(1500\)	\(1000\)

Represent the above data by a bar graph.
Ans: The required bar graph is shown below.

Q.3. The following table shows a state government’s expenditure in the year 2010.

Category	Security and external relations	Social development	Economic development	Governement administration
Amount (in crore \(₹\)	\(14,311\)	\(15,400\)	\(1,914\)	\(1,130\)

Construct a pie chart to represent the data.
Ans: The area of the sector of the circle is proportional to the angle subtended at the centre of the sector. Thus, we have to calculate the angle of each sector first.

Category	Amount (in crore \(₹\))	Angle of sector
Security and external relations	\(14,311\)	\(\frac{{14,311}}{{32,755}} \times {360^ \circ } = {157.3^ \circ }\)
Social development	\(15,400\)	\(\frac{{15,400}}{{32,755}} \times {360^ \circ } = {169.3^ \circ }\)
Economic development	\(19.14\)	\(\frac{{19,14}}{{32,755}} \times {360^ \circ } = {21.0^ \circ }\)
Government administration	\(1,130\)	\(\frac{{1.130}}{{32,755}} \times {360^ \circ } = {12.4^ \circ }\)
Total government expenditure	\(14,311 + 15,400 + 1,914 + 1,130 = 32,755\)

 We can draw the pie chart and label it as shown below.

Q.4. The following table shows the heights of 50 students.

Height (in \({\text{cm}}\))	Frequency
\(150 – 155\)	\(7\)
\(155 – 160\)	\(12\)
\(160 – 165\)	\(18\)
\(165 – 170\)	\(10\)
\(170 – 175\)	\(3\)

a) Represent the data using a histogram
Ans: a) The histogram is as shown below.

Q.5. The following table gives the distribution of students of two sections according to their marks.

Represent marks of the students of both the sections on the same graph by two frequency polygon.
Ans: We find the class marks and prepare a new table as shown below.

Class	Class Marks	Section A	Section B
\(0 – 10\)	\(5\)	\(3\)	\(5\)
\(10 – 20\)	\(15\)	\(9\)	\(9\)
\(20 – 30\)	\(25\)	\(17\)	\(15\)
\(30 – 40\)	\(35\)	\(12\)	\(10\)
\(40 – 50\)	\(45\)	\(9\)	\(1\)

• Steps to draw frequency polygon:
• Take \(1\,{\text{cm}}\) on the \(x\)-axis \( = 5\) marks and \(1\,{\text{cm}}\) on the \(y\)-axis \( = 5\) students
• For section \(A\)
• Plot the points \(\left({5,3} \right),\left({15,9} \right),\left({25,17} \right),\left({35,12} \right)\) and \(\left({45,9} \right).\)
• Join the points by thick line segments.
• Join the first endpoint with the midpoint of the class \(\left({\left({ – 10} \right) – 0} \right)\) with zero frequency, and join the other endpoint with the midpoint of class \(50 – 60\) with zero frequency. Thick line segments show the required frequency polygon in the below-given figure.
• For section \(B\)
• Plot the points \(\left({5,5} \right),\left({15,19}\right),\left({25,15} \right),\left({35,10} \right)\) and \(\left({45,1} \right).\)
• Join the points with dotted line segments.
• Join the first endpoint with the midpoint of the class \(\left({\left({ – 10} \right) – 0} \right)\) with zero frequency, and join the other endpoint with the midpoint of class \(50 – 60\) with zero frequency. The dotted line segments show the required frequency polygon in the below-given figure.

Summary

In this article, we took a quick view of the graphs, and then we listed out some types of graphs, and later we learnt the applications of graphs in detail. Nowadays, graphs are used in every field, whether the medical field, biotechnology, or artificial intelligence. Lastly, we solved some examples based on graphs to strengthen our grip on the concept of applications of graphs.

FAQs

Q.1. What are the applications of a graph?
Ans: Below given are a few fields where the application of graphs is beneficial.
1. Graphs in quantum field theory
2. Semantic networks
3. Graphs in compilers
4. Graphs in epidemiology
5. To compare the data
6. Network traffic packet graph

Q.2. What is the use of graphs in mathematics?
Ans: Below given are some uses of graphs in mathematics.
1. Bar graphs are helpful to represent when the data are in categories.
2. A histogram is used to represent grouped data with class intervals.
3. A pie chart helps show the relative size of individual categories to the total.

Q.3. What are the different types of graphs?
Ans: The pictorial representation of data or information is called a graph. There are various ways of representing numerical data graphically.
1. Bar graph
2. Histograms
3. Pie graph
4. Frequency polygon graph
5. Line graph
6. Linear graph

Q.4. What are real-life applications of graphs?
Ans: Graphs represent networks of communication. The shortest path in a road or network is determined using graphs. Various locations are represented as vertices or nodes, while highways are represented as edges, with graph theory being utilised to calculate the shortest path between two nodes in Google Maps.

Q.5. What are the advantages of a graph?
Ans: The information in numerical data can be easily understood if we represent it in diagrams or graphs.  It is well said that one picture is better than a thousand words. Usually, comparisons among the individual are best shown through graphs.