Graph Formula

Graphs are mathematical constructs used to represent pairwise relationships between entities. A graph is composed of vertices (or nodes) and edges that connect these vertices. The formulas related to graphs are employed to characterize their attributes and dynamics. In this article, students will learn in detail about graph formula, solved examples based on it etc,. in detail.

What is Graph Formula?

In mathematics and computer science, a graph formula is a mathematical expression or a set of rules that define the properties and structure of a graph. Graphs are utilized to represent relationships between objects, and different formulas are applied to analyze and comprehend these relationships.

Definition of Graph

A graph is a mathematical framework designed to model pairwise relationships between objects. It comprises a set of vertices (or nodes) and a set of edges (or links) that connect pairs of these vertices. Formally, a graph G is represented as an ordered pair (V,E), where V denotes the set of vertices and E denotes the set of edges. Each edge in E signifies a connection or relationship between two vertices in V. Graphs are extensively used across various disciplines, including computer science, mathematics, and engineering, to depict and analyze networks and relational structures.

Graph Notations

Vertex Set: V (G) represents the collection of vertices in the graph G.

Edge Set: E (G) represents the collection of edges in the graph G.

Degree of a Vertex: This refers to the number of edges connected to a vertex. In the context of a directed graph, this is further categorized into in-degree (the number of edges coming into the vertex) and out-degree (the number of edges going out from the vertex).

Basic Terminology

Path: A sequence of vertices where each consecutive pair is connected by an edge.

Cycle: A path that begins and ends at the same vertex, with no other vertices or edges repeated.

Adjacency Matrix: A square matrix A used to represent a finite graph. For a graph with n vertices, A is an n×n matrix where:

• A[i][j] = 1 if there is an edge from vertex i to vertex j.
• A[i][j]=0 otherwise.

Incidence Matrix: A matrix representing the relationship between vertices and edges. For a graph with n vertices and m edges, the incidence matrix B is an n×m matrix where:

• B[i][j]=1 if vertex i is incident to edge j.
• B[i][j]=0 otherwise.

Eulerian Path: A path that traverses every edge exactly once.

Eulerian Circuit: A circuit that traverses every edge exactly once and returns to the starting vertex.

Slope Intercept Formula for Graph

The slope-intercept formula for a straight line passing through two points, (a₁, b₁) and (a₂, b₂), is given by y=mx+b.

The relationship between the two points, used to plot the line on the graph, is described by the Graph Formula, also known as the slope-intercept form of the straight-line equation. The Graph Formula simplifies the process of plotting graphs. It is derived using the coordinates of the two points on the line. The Graph Formula is written as y=mx+b, where mmm is the slope. The slope mmm is also known as “rise over run,” indicating how many units the line moves up or down for each unit it moves horizontally. In the Graph Formula, bbb is the y-int/ercept, indicating the point where the line crosses the y-axis.

To summarize, assuming the two points are (a₁, b₁) and (a2, b₂), the slope-intercept form of the straight line can be calculated using the following Graph Formula:

y = mx+b

Where:

• m is the slope, calculated as b2−b1/a2−a1​​.
• b is the y-intercept.

Types of Graph

A graph functions as a mathematical representation of networks, designed to visually convey mathematical relationships for better understanding. There are several types of graph formats, including:

Bar Graph

A bar graph, also referred to as a bar chart, visually represents data using rectangular bars or columns. Each bar corresponds to a specific category or data point, with the length or height of the bar proportional to the value it signifies. Bar graphs are often used to compare different categories or to illustrate data changes over time. They offer a straightforward visual method for quickly comprehending and analyzing numerical data and trends.

Pie Graph

A pie graph, also called a pie chart, is a circular visual representation used to depict the proportions or percentages of data within a whole. The circle represents the entirety of the data set, divided into individual “slices” or sectors, each corresponding to a specific category or data point. The size of each slice is relative to the quantity it represents compared to the whole dataset.

Pie graphs are effective for displaying the distribution of categorical data and demonstrating how different segments contribute to the overall composition. They provide a visual means to quickly understand the relative sizes of various components within a dataset. The angles of the slices are calculated based on the proportions they represent, facilitating easy comparison of the magnitudes of different categories at a glance.

Line Graph

A line graph, also known as a line chart, is a graphical representation that presents data points as markers connected by lines. This graph is commonly utilized to illustrate the correlation between two or more variables and to depict their fluctuations over a continuous or discrete period. Typically, in a line graph, the x-axis represents the independent variable, often time, while the y-axis represents the dependent variable. Data points are plotted at specific positions on the graph, and lines are drawn to link these points. These lines offer a visual representation of the trend, pattern, or variation in the data across the specified range. Line graphs are valuable tools for visualizing changes and trends in data over time or other ordered sequences.

Histogram Graph

A histogram graph visually represents data using rectangular bars of different heights, where each bar represents a particular range of quantitative values. Unlike a bar graph, which is used for categorical variables, a histogram focuses on displaying the distribution of quantitative data. Each bar in a histogram corresponds to a specific range or interval of values along the horizontal axis, known as bins or buckets. The height of each bar indicates the frequency or count of data points falling within that range. Typically, the bars in a histogram are adjacent and touch each other to emphasize the continuous nature of the data.

Scatter Diagrams

A scatter diagram, commonly known as a scatter plot, is a visual representation used to depict the relationship between two variables. It utilizes Cartesian coordinates, with one variable plotted on the horizontal axis (x-axis) and the other variable plotted on the vertical axis (y-axis). In a scatter plot, individual data points are represented by dots or markers on the graph, with each dot corresponding to a specific combination of values for the two variables. By examining the pattern formed by these data points, one can assess the correlation or relationship between the variables.

Solved Examples on Graph Formula

Example 1: Determine the slope between the points (2, 4) and (6, 10).

Solution:

Using the slope formula:

Slope (m)=(b2−b1)/(a2−a1)

​Substitute the given coordinates:

Slope (m)=(10−4)/(6−2)

Slope (m)=1.5

Therefore, the slope between the points (2, 4) and (6, 10) is 1.5.

Example 2: Determine the slope between the points (-3, 5) and (2, -1).

Solution:

Using the slope formula:

Slope (m)=(b2−b1)/(a2−a1)

​Substitute the given coordinates:

Slope (m)=(-1−5)/(2−(-3)

Slope (m)=−1.2

Therefore, the slope between the points (-3, 5) and (2, -1) is -1.2.

Example 3: Find the slope between the points (-1, 3) and (-4, 7).

Solution:

Using the slope formula:

Slope (m)=(b2−b1)/(a2−a1)

​Substitute the given coordinates:

Slope (m)=(7−3)/(−4−(−1)

​Slope (m)=−4/3