Types of Graphs in Data Structure & Applications

Graphs in data structures and applications are nonlinear frameworks comprising a finite number of vertices (nodes) and the edges that connect them. Professionals often use different types of graphs to address real-world problems, representing the problem area as a particular network. For example, a graph can represent a user as a node in a telephone network, with the link between users through a phone call serving as the edge.

If you’re a data science professional, you must know the different data structure trees and graphs to solve complex problems and optimize algorithms. Knowledge of these graphs also enables you to analyze relationships between different data points in real-life applications. This makes you a valuable candidate for jobs in data science, software engineering, machine learning, and other tech-driven roles. In this guide, we’ll explore the types of graphs in data structures and applications in detail. 

Why Are So Many Graphs Required? 

A graph data structure represents multiple computer networks, where devices are the vertices and communication links are the edges. Moreover, search engines like Google also use graphs to index the web. You must understand what are data structures and algorithms to draft graphs across systems and applications effectively.

The following points further highlight the significance of graphs in data structures and applications:

Diverse Problem-Solving Needs

Different types of graphs in data structures cater to specific real-world problems. For instance, 

• Social networks use undirected graphs, where users are nodes and friendships are edges.
• Traffic flow is optimized with directed graphs, where intersections are vertices and roads are edges.
• Organizational hierarchies can be represented by tree graphs to show relationships between employees and departments.

These varied graph structures provide efficient solutions to complex, domain-specific challenges across any data science process.

Algorithm Optimization 

The choice of graph type impacts the efficiency of algorithms in search, traversal, connectivity analysis, and other related tasks. For example, depth-first search (DFS) is more effective on tree graphs, while breadth-first search (BFS) is ideal for unweighted graphs. 

Weighted graphs require Dijkstra's algorithm for shortest-path calculations, while directed graphs enhance flow analysis or cycle detection. A solid understanding of graph types, algorithm complexity, and data structure helps you select optimal algorithms for specific tasks, improving performance and reducing computational overhead.

Multiple Use Cases 

Graphs enable professionals to solve practical issues across industries by representing interconnected data visually and systematically. This also allows professionals and organizations to develop streamlined solutions for complex problems while understanding concepts such as data vs. information, interconnectivity, and modeling structures.

Various types of graphs in data structures have multiple real-world applications. For example:

• Bipartite graphs are used in matchmaking systems or job assignment problems, where edges connect two distinct sets.
• Planar graphs help in network design processes, ensuring no edges cross when laying out connections in a physical space.

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Foundational Types of Graphs

Foundational types of graphs in data structures serve as basic visual representations of data. These graphs allow for easy comparison and analysis of trends across different categories or variables, depending on the presented data. You can easily understand what is the future scope of computer science and solve different problems and real-world applications with these graph types.

Directed vs. Undirected Graphs

Directed and undirected graphs represent two primary types of relationships between nodes. A directed graph contains edges that have a specific direction. Each edge is represented as an ordered pair of vertices, indicating movement from a source vertex to a destination vertex. Conversely, an undirected graph has edges with no direction, meaning the relationship between nodes is bidirectional.

The following table highlights the primary differences between directed and undirected graphs:

Weighted vs. Unweighted Graphs

Graphs can also be categorized based on whether the edges have weights. A weighted graph assigns a weight or cost to each edge, representing distance, time, or another measurable factor between nodes. Conversely, an unweighted graph treats all edges equally, with no associated weights.

Here’s a comparison table highlighting the primary differences:

Learn more about the major differences between data graphs by pursuing upGrad’s Big Data courses. 

Connectivity-based Classifications

Connectivity is a fundamental concept in graph theory. It refers to the minimum number of nodes or edges that must be removed to isolate the remaining elements into separate subgraphs. Connectivity-based classifications of graphs are closely related to computer networking basics and flow problems, which involve determining the maximum flow of data or resources from a source node to a destination node.

Connected vs. Disconnected Graphs

Graphs are often classified based on whether all vertices are reachable from any other vertex. A connected graph contains a path between every pair of vertices.

A common example is a social network, where each user (node) can reach another user through direct or indirect connections. Most routing algorithms require a connected graph to determine the shortest path.

Conversely, a disconnected graph has at least one pair of vertices that cannot be reached from each other. For instance, a power grid with isolated substations forms a disconnected graph in which some substations do not receive electricity.

The primary differences between connected and disconnected graphs are outlined below:

Cyclic vs. Acyclic Graphs

Graphs can also be classified based on the presence of cycles, which are paths that start and end at the same vertex without repeating an edge.

A cyclic graph contains at least one cycle. A common example is a road network where routes form loops. An acyclic graph, on the other hand, contains no cycles. A family tree data structure is a typical example, where edges represent parent-child relationships.

The differences between cyclic and acyclic graphs are outlined in the table below:

The classification of graphs as cyclic or acyclic plays a significant role in algorithm design and problem-solving in real-world applications.

Specialized Graph Structures

Specialized graph structures possess unique properties and serve specific purposes. They are widely used in real-life applications, such as social networks, routing algorithms, and recommendation systems.

Below is an overview of different types of specialized graph structures:

Complete Graphs

A complete graph is a graph in which every pair of distinct vertices is connected by a unique edge. In other words, each node is directly linked to every other node. This structure is useful in scenarios such as network design, where every device must be directly connected to all others for maximum communication efficiency. It is also applicable to problems involving full connectivity and optimization.

In network design, a complete graph allows every device to be directly connected. This practical application helps prevent delays and facilitates efficient communication channels for businesses.

Bipartite Graphs

A bipartite graph consists of two distinct sets of vertices, with edges only running between vertices of different sets, never within the same set. This graph structure is particularly useful in matching problems, such as job assignments or task allocation. In this case, one set represents workers, and the other set represents tasks.

A few real-world examples of bipartite graphs include:

• Matching Problems: One set could represent job roles, and the other set could represent candidates in job assignments. The edges indicate the suitability of candidates for specific jobs.
• Recommendation Systems: Bipartite graphs are also used in collaborative filtering, where one set represents users, and the other represents recommended items. 

Subgraphs

A subgraph is formed by selecting a subset of vertices and edges from a larger graph. It retains the structure of the original graph but focuses on a specific part of it. Subgraphs are often used to analyze particular segments or components of a larger network.

For instance:

• A subgraph could represent a group of friends or followers within a larger user base on social media platforms.
• It could also consist of a subset of web pages used to categorize valuable insights into smaller portions of the web.

Multigraphs

A multigraph is a graph where multiple edges can exist between two nodes, meaning there is no restriction on the number of edges between any pair of vertices. This structure is useful in scenarios where multiple types of connections exist between two entities, such as different routes between two locations in a transportation network.

For example, multiple bus routes (edges) could connect two cities (nodes) in a transportation system, representing different travel options for users.

Null Graphs

A null graph is one with no vertices and no edges. This is the most basic graph form, where no connections exist between any nodes. Null graphs often serve as base cases in graph theory and can be used in algorithms that require an initial graph with no connections.

A common visual representation of a null graph with three vertices would appear as three isolated points with no edges between them.

Regular Graphs

A regular graph has the same number of edges, meaning each vertex has the same degree. This type of graph is particularly useful for modeling situations where every entity has the same number of interactions or connections, such as in uniform network setups.

Key features: 

• Every vertex has the same number of edges.
• Each person in a network might have the same number of friends or connections in a regular graph. 

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Hierarchical Graph Structures

A hierarchical graph structure, also known as an H-graph, organizes data at different levels of detail. It represents information at varying levels of abstraction.

Trees

A tree is a special type of hierarchical graph in which nodes are connected in a parent-child relationship without forming cycles. Each tree has a single root node, and every node (except the root) has exactly one parent. Trees are widely used in applications such as file systems, organizational charts, and search algorithms. You can explore more about their structure and applications in a Trees in Data Structure tutorial.

Properties of Trees:

• Root Node: The topmost node with no parent.
• Edges: Connections between nodes.
• No Cycles: There are no loops or repeated connections.
• Height: The longest path from the root to a leaf node.

Here are some of the use cases of tree graphs that you must know about: 

Forests

A forest is a collection of disjoint trees, meaning it consists of multiple unconnected tree structures. Each component follows tree properties, but there is no single root node linking all trees. Forest data structures are useful in scenarios where hierarchical structures exist independently within the same system.

Common Examples and Applications of Forests:

• Database Indexing: Used in B-trees and other indexing techniques for efficient data management.
• Connected Components in Graphs: When edges are removed from a graph, the remaining disconnected trees form a forest.
• Organization of Web Pages: Different website categories can be represented as separate trees within a forest.

Geometric Graph Structures

A geometric graph structure is a type of graph in which nodes are positioned in a geometric space, with edges defined based on their spatial relationships. These relationships are typically represented as straight-line segments, and each vertex has geometric coordinates.

Planar Graphs

A planar graph can be drawn on a plane without any edges crossing each other. This property is essential for applications such as circuit board design, transportation networks, and geographical mapping. Avoiding edge overlaps enhances efficiency and clarity in these graph types.

Here are the rules associated with planar graphs: 

• A graph with ≤ 4 vertices is always planar.
• K₄ (a complete graph with 4 nodes) is planar, but K₅ (a complete graph with 5 nodes) and K₃,₃ (a bipartite complete graph with 6 nodes) are non-planar.
• A graph is non-planar if it contains a subgraph homeomorphic to K₅ or K₃,₃.

Here are some practical examples of planar graph diagrams: 

• Road Networks: Planar graphs help model road layouts to ensure intersections occur only at designated points.
• Printed Circuit Boards (PCBs): Used in routing circuits without overlapping connections in electronic designs.
• Geographic Mapping: Maps can be represented as planar graphs, where edges define borders between regions.

Dense vs. Sparse Graphs

Graphs can be categorized as dense or sparse based on the number of edges relative to the number of vertices. A dense graph has a high number of edges, similar to a social network where most users are connected to several other people.

Conversely, a sparse graph has relatively few edges compared to the total possible connections. A common example is a road network, where cities (nodes) are connected to only a few nearby areas rather than all possible locations.

Here is a comparison table that showcases the differences between dense and sparse graphs: 

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How to Implement All Types of Graphs

Adjacency matrices and lists are the two most common ways to implement graphs. However, other alternatives are available depending on the graph type and application. The following sections outline different graph representation methods and tools used for implementation.

Representation Methods

Choosing the right representation method improves graph performance and storage efficiency. Each approach offers different advantages based on the graph’s density, size, and intended use.

Below are the common methods for representing graphs in data structures:

• Adjacency Matrix: Uses a 2D array of V × V vertices, where each row and column represent a vertex. This method is straightforward but can be inefficient for sparse graphs due to high memory usage.
• Adjacency List: This method represents a graph as an array (A) of linked lists, where each vertex maintains a list of its adjacent vertices. It is memory-efficient for sparse graphs.
• Incidence Matrix: Uses a matrix where rows represent vertices and columns represent edges, with entries indicating vertex-edge incidences. This is useful for certain mathematical applications and graph algorithms.

Read More: 

• Adjacency Matrix Tutorial
• Adjacency Lists in Data Structure Tutorial 

Tools and Libraries

Several programming tools and libraries simplify graph implementation by providing built-in functionalities for graph creation, manipulation, and analysis. These can be considered among the best applications of Big Data in the real-world that is omnipresent across industries. 

Below are some of the most widely used tools and libraries:

• NetworkX (Python): Provides a simple and flexible way to create, manipulate, and analyze complex networks. It is ideal for small to medium-sized graphs.
• GraphX (Apache Spark): IProvides a simple and flexible way to create, manipulate, and analyze complex networks. It is ideal for small to medium-sized graphs.
• Neo4j (Graph Database): A highly scalable graph database that supports Atomicity, Consistency, Isolation, and Durability (ACID) transactions, making it ideal for enterprise-level graph-based applications.
• Gephi (Visualization): A tool for visualizing graphs and exploring their structures through graphical interfaces, often used for social network analysis and data exploration.

Real-World Use Cases

If you’re an aspiring data scientist or engineer, you must understand how to apply graph representations in real-world scenarios. This makes the implementation process for data visualization more intuitive.

Example 1: Social Network Analysis

import networkx as nx
G = nx.Graph()
G.add_edges_from([("Alice", "Bob"), ("Alice", "Carol"), ("Bob", "David")])
nx.draw(G, with_labels=True)

Example 2: Route Optimization with Adjacency Matrix

import numpy as np
adj_matrix = np.array([
    [0, 1, 0, 1],
    [1, 0, 1, 0],
    [0, 1, 0, 1],
    [1, 0, 1, 0]
])
print("Adjacency Matrix:\n", adj_matrix)

Example 3: Supply Chain Graph using Incidence Matrix

import numpy as np
incidence_matrix = np.array([
    [1, 0, 0],
    [1, 1, 0],
    [0, 1, 1],
    [0, 0, 1]
])
print("Incidence Matrix:\n", incidence_matrix)

Bottom Line

Graphs have wide-ranging applications in both practical and theoretical domains. The best Big Data applications in the real world function with the assistance of various types of graphs in data structures. They usually represent the outcome or result of solving real-world problems.

Applying graph theory to a specific domain or application typically involves using graphs to represent relationships, connections, or patterns between entities. If you’re a data scientist or data engineer, you can explore different graph types through professional courses available at upGrad. You can enroll in the platform’s Big Data courses now.

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