Introduction to Graphs – Data Structure and Algorithm Tutorials

Graph data structures are a powerful tool for representing and analyzing complex relationships between objects or entities. They are particularly useful in fields such as social network analysis, recommendation systems, and computer networks. In the field of sports data science, graph data structures can be used to analyze and understand the dynamics of team performance and player interactions on the field.

What is a Graph?

Graph is a non-linear data structure consisting of vertices and edges. The vertices are sometimes also referred to as nodes and the edges are lines or arcs that connect any two nodes in the graph. More formally a Graph is composed of a set of vertices( V ) and a set of edges( E ). The graph is denoted by G(V, E).

Imagine a game of football as a web of connections, where players are the nodes and their interactions on the field are the edges. This web of connections is exactly what a graph data structure represents, and it’s the key to unlocking insights into team performance and player dynamics in sports.

Components of a Graph:

• Vertices: Vertices are the fundamental units of the graph. Sometimes, vertices are also known as vertex or nodes. Every node/vertex can be labeled or unlabelled.
• Edges: Edges are drawn or used to connect two nodes of the graph. It can be ordered pair of nodes in a directed graph. Edges can connect any two nodes in any possible way. There are no rules. Sometimes, edges are also known as arcs. Every edge can be labelled/unlabelled.

Types Of Graph

1. Null Graph

A graph is known as a null graph if there are no edges in the graph.

2. Trivial Graph

Graph having only a single vertex, it is also the smallest graph possible. 

3. Undirected Graph

A graph in which edges do not have any direction. That is the nodes are unordered pairs in the definition of every edge. 

4. Directed Graph

A graph in which edge has direction. That is the nodes are ordered pairs in the definition of every edge.

5. Connected Graph

The graph in which from one node we can visit any other node in the graph is known as a connected graph. 

6. Disconnected Graph

The graph in which at least one node is not reachable from a node is known as a disconnected graph.

7. Regular Graph

The graph in which the degree of every vertex is equal to K is called K regular graph.

8. Complete Graph

The graph in which from each node there is an edge to each other node.

9. Cycle Graph

The graph in which the graph is a cycle in itself, the degree of each vertex is 2. 

10. Cyclic Graph

A graph containing at least one cycle is known as a Cyclic graph.

11. Directed Acyclic Graph

A Directed Graph that does not contain any cycle. 

12. Bipartite Graph

A graph in which vertex can be divided into two sets such that vertex in each set does not contain any edge between them.

13. Weighted Graph

• A graph in which the edges are already specified with suitable weight is known as a weighted graph. 
•  Weighted graphs can be further classified as directed weighted graphs and undirected weighted graphs. 

Representation of Graphs:

There are two ways to store a graph:

• Adjacency Matrix
• Adjacency List

Adjacency Matrix Representation of Graph:

In this method, the graph is stored in the form of the 2D matrix where rows and columns denote vertices. Each entry in the matrix represents the weight of the edge between those vertices. 

Below is the implementation of Graphs represented using Adjacency Matrix:

C++

#include <iostream>
#include <vector>

using namespace std;

vector<vector<int> >
createAdjacencyMatrix(vector<vector<int> >& graph,
                      int numVertices)
{
    // Initialize the adjacency matrix with zeros
    vector<vector<int> > adjMatrix(
        numVertices, vector<int>(numVertices, 0));

    // Fill the adjacency matrix based on the edges in the
    // graph
    for (int i = 0; i < numVertices; ++i) {
        for (int j = 0; j < numVertices; ++j) {
            if (graph[i][j] == 1) {
                adjMatrix[i][j] = 1;
                adjMatrix[j][i]
                    = 1; // For undirected graph, set
                         // symmetric entries
            }
        }
    }

    return adjMatrix;
}

int main()
{
    // The indices represent the vertices, and the values
    // are lists of neighboring vertices
    vector<vector<int> > graph = { { 0, 1, 0, 0 },
                                   { 1, 0, 1, 0 },
                                   { 0, 1, 0, 1 },
                                   { 0, 0, 1, 0 } };

    int numVertices = graph.size();

    // Create the adjacency matrix
    vector<vector<int> > adjMatrix
        = createAdjacencyMatrix(graph, numVertices);

    // Print the adjacency matrix
    for (int i = 0; i < numVertices; ++i) {
        for (int j = 0; j < numVertices; ++j) {
            cout << adjMatrix[i][j] << " ";
        }
        cout << endl;
    }

    return 0;
}

Java

import java.util.ArrayList;
import java.util.Arrays;

public class AdjacencyMatrix {
    public static int[][] createAdjacencyMatrix(
        ArrayList<ArrayList<Integer> > graph,
        int numVertices)
    {
        // Initialize the adjacency matrix with zeros
        int[][] adjMatrix
            = new int[numVertices][numVertices];

        // Fill the adjacency matrix based on the edges in
        // the graph
        for (int i = 0; i < numVertices; ++i) {
            for (int j = 0; j < numVertices; ++j) {
                if (graph.get(i).get(j) == 1) {
                    adjMatrix[i][j] = 1;
                    adjMatrix[j][i]
                        = 1; // For undirected graph, set
                             // symmetric entries
                }
            }
        }

        return adjMatrix;
    }

    public static void main(String[] args)
    {
        // The indices represent the vertices, and the
        // values are lists of neighboring vertices
        ArrayList<ArrayList<Integer> > graph
            = new ArrayList<>();
        graph.add(
            new ArrayList<>(Arrays.asList(0, 1, 0, 0)));
        graph.add(
            new ArrayList<>(Arrays.asList(1, 0, 1, 0)));
        graph.add(
            new ArrayList<>(Arrays.asList(0, 1, 0, 1)));
        graph.add(
            new ArrayList<>(Arrays.asList(0, 0, 1, 0)));

        int numVertices = graph.size();

        // Create the adjacency matrix
        int[][] adjMatrix
            = createAdjacencyMatrix(graph, numVertices);

        // Print the adjacency matrix
        for (int i = 0; i < numVertices; ++i) {
            for (int j = 0; j < numVertices; ++j) {
                System.out.print(adjMatrix[i][j] + " ");
            }
            System.out.println();
        }
    }
}

// This code is contributed by shivamgupta310570

Python

def create_adjacency_matrix(graph):
    # Get the number of vertices in the graph
    num_vertices = len(graph)

    # Initialize the adjacency matrix with zeros
    adj_matrix = [[0] * num_vertices for _ in range(num_vertices)]

    # Fill the adjacency matrix based on the edges in the graph
    for i in range(num_vertices):
        for j in range(num_vertices):
            if graph[i][j] == 1:
                adj_matrix[i][j] = 1
                # For undirected graph, set symmetric entries
                adj_matrix[j][i] = 1

    return adj_matrix


# The indices represent the vertices, and the values are lists of neighboring vertices
graph = [
    [0, 1, 0, 0],
    [1, 0, 1, 0],
    [0, 1, 0, 1],
    [0, 0, 1, 0]
]

# Create the adjacency matrix
adj_matrix = create_adjacency_matrix(graph)

# Print the adjacency matrix
for row in adj_matrix:
    print(' '.join(map(str, row)))

# This code is contributed by shivamgupta0987654321

JavaScript

function createAdjacencyMatrix(graph) {
    const numVertices = graph.length;

    // Initialize the adjacency matrix with zeros
    const adjMatrix = Array.from(Array(numVertices), () => Array(numVertices).fill(0));

    // Fill the adjacency matrix based on the edges in the graph
    for (let i = 0; i < numVertices; ++i) {
        for (let j = 0; j < numVertices; ++j) {
            if (graph[i][j] === 1) {
                adjMatrix[i][j] = 1;
                adjMatrix[j][i] = 1; // For undirected graph, set symmetric entries
            }
        }
    }

    return adjMatrix;
}

// The indices represent the vertices, and the values are lists of neighboring vertices
const graph = [
    [0, 1, 0, 0],
    [1, 0, 1, 0],
    [0, 1, 0, 1],
    [0, 0, 1, 0]
];

// Create the adjacency matrix
const adjMatrix = createAdjacencyMatrix(graph);

// Print the adjacency matrix
for (let i = 0; i < adjMatrix.length; ++i) {
    console.log(adjMatrix[i].join(' '));
}

Adjacency List Representation of Graph:

This graph is represented as a collection of linked lists. There is an array of pointer which points to the edges connected to that vertex. 

Below is the implementation of Graphs represented using Adjacency List:

C++

#include <bits/stdc++.h>

using namespace std;

vector<vector<int>> createAdjacencyList(vector<vector<int> >& edges, int numVertices)
{
    // Initialize the adjacency list 
    vector<vector<int> > adjList(numVertices);

    // Fill the adjacency list based on the edges in the
    // graph
    for (int i=0; i < edges.size(); i++) {
        int u = edges[i][0];
          int v = edges[i][1];
          // Since the graph is undirected, therefore we push the edges in both the directions
          adjList[u].push_back(v);
          adjList[v].push_back(u);
    }

    return adjList;
}

int main()
{
      // Undirected Graph of 4 nodes
      int numVertices = 4;
      vector<vector<int>> edges = {{0, 1}, {0, 2}, {1, 2}, {2, 3}, {3, 1}};
  
    // Create the adjacency List
    vector<vector<int>> adjList = createAdjacencyList(edges, numVertices);

    // Print the adjacency List
    for (int i = 0; i < numVertices; ++i) {
          cout << i << " -> ";
        for (int j = 0; j < adjList[i].size(); ++j) {
            cout << adjList[i][j] << " ";
        }
        cout << endl;
    }

    return 0;
}

Java

import java.util.ArrayList;
import java.util.List;

public class Main {
    public static List<List<Integer> >
    createAdjacencyList(List<List<Integer> > edges,
                        int numVertices)
    {
        // Initialize the adjacency list
        List<List<Integer> > adjList
            = new ArrayList<>(numVertices);
        for (int i = 0; i < numVertices; i++) {
            adjList.add(new ArrayList<>());
        }

        // Fill the adjacency list based on the edges in the
        // graph
        for (List<Integer> edge : edges) {
            int u = edge.get(0);
            int v = edge.get(1);
            // Since the graph is undirected, therefore we
            // push the edges in both the directions
            adjList.get(u).add(v);
            adjList.get(v).add(u);
        }

        return adjList;
    }

    public static void main(String[] args)
    {
        // Undirected Graph of 4 nodes
        int numVertices = 4;
        List<List<Integer> > edges = new ArrayList<>();
        edges.add(List.of(0, 1));
        edges.add(List.of(0, 2));
        edges.add(List.of(1, 2));
        edges.add(List.of(2, 3));
        edges.add(List.of(3, 1));

        // Create the adjacency list
        List<List<Integer> > adjList
            = createAdjacencyList(edges, numVertices);

        // Print the adjacency list
        for (int i = 0; i < numVertices; ++i) {
            System.out.print(i + " -> ");
            for (int j = 0; j < adjList.get(i).size();
                 ++j) {
                System.out.print(adjList.get(i).get(j)
                                 + " ");
            }
            System.out.println();
        }
    }
}

Python

def create_adjacency_list(edges, num_vertices):
    # Initialize the adjacency list
    adj_list = [[] for _ in range(num_vertices)]

    # Fill the adjacency list based on the edges in the graph
    for u, v in edges:
        # Since the graph is undirected, push the edges in both directions
        adj_list[u].append(v)
        adj_list[v].append(u)

    return adj_list

if __name__ == "__main__":
    # Undirected Graph of 4 nodes
    num_vertices = 4
    edges = [(0, 1), (0, 2), (1, 2), (2, 3), (3, 1)]

    # Create the adjacency list
    adj_list = create_adjacency_list(edges, num_vertices)

    # Print the adjacency list
    for i in range(num_vertices):
        print(f"{i} -> {' '.join(map(str, adj_list[i]))}")

C#

using System;
using System.Collections.Generic;

class MainClass {
    public static List<List<int> >
    CreateAdjacencyList(List<List<int> > edges,
                        int numVertices)
    {
        // Initialize the adjacency list
        List<List<int> > adjList
            = new List<List<int> >(numVertices);
        for (int i = 0; i < numVertices; i++) {
            adjList.Add(new List<int>());
        }

        // Fill the adjacency list based on the edges in the
        // graph
        foreach(List<int> edge in edges)
        {
            int u = edge[0];
            int v = edge[1];
            // Since the graph is undirected, push the edges
            // in both directions
            adjList[u].Add(v);
            adjList[v].Add(u);
        }

        return adjList;
    }

    public static void Main(string[] args)
    {
        // Undirected Graph of 4 nodes
        int numVertices = 4;
        List<List<int> > edges = new List<List<int> >{
            new List<int>{ 0, 1 }, new List<int>{ 0, 2 },
            new List<int>{ 1, 2 }, new List<int>{ 2, 3 },
            new List<int>{ 3, 1 }
        };

        // Create the adjacency list
        List<List<int> > adjList
            = CreateAdjacencyList(edges, numVertices);

        // Print the adjacency list
        for (int i = 0; i < numVertices; i++) {
            Console.Write($"{i} -> ");
            foreach(int vertex in adjList[i])
            {
                Console.Write($"{vertex} ");
            }
            Console.WriteLine();
        }
    }
}

JavaScript

function createAdjacencyList(edges, numVertices) {
    // Initialize the adjacency list
    const adjList = new Array(numVertices).fill(null).map(() => []);

    // Fill the adjacency list based on the edges in the graph
    edges.forEach(([u, v]) => {
        // Since the graph is undirected, push the edges in both directions
        adjList[u].push(v);
        adjList[v].push(u);
    });

    return adjList;
}

// Undirected Graph of 4 nodes
const numVertices = 4;
const edges = [[0, 1], [0, 2], [1, 2], [2, 3], [3, 1]];

// Create the adjacency list
const adjList = createAdjacencyList(edges, numVertices);

// Print the adjacency list
for (let i = 0; i < numVertices; i++) {
    console.log(`${i} -> ${adjList[i].join(' ')}`);
}

Output

0 -> 1 2 
1 -> 0 2 3 
2 -> 0 1 3 
3 -> 2 1 

Comparison between Adjacency Matrix and Adjacency List

When the graph contains a large number of edges then it is good to store it as a matrix because only some entries in the matrix will be empty. An algorithm such as Prim’s and Dijkstra adjacency matrix is used to have less complexity.

Action	Adjacency Matrix	Adjacency List
Adding Edge	O(1)	O(1)
Removing an edge	O(1)	O(N)
Initializing	O(N*N)	O(N)

Basic Operations on Graphs

Below are the basic operations on the graph:

• Insertion or Deletion of Nodes in the graph
  • Add and Remove vertex in Adjacency List representation of Graph
  • Add and Remove vertex in Adjacency Matrix representation of Graph
• Insertion or Deletion of Edges in the graph
  • Add and Remove Edge in Adjacency List representation of a Graph
  • Add and Remove Edge in Adjacency Matrix representation of a Graph
• Searching on Graphs – Search an entity in the graph.
• Traversal of Graphs – Traversing all the nodes in the graph.

Difference between Tree and Graph:

Trees are the restricted types of graphs, just with some more rules. Every tree will always be a graph but not all graphs will be trees. Linked List, Trees, and Heaps all are special cases of graphs. 

Real-Life Applications of Graph:

Graphs have numerous real-life applications across various fields. Some of them are listed below:

• Graphs are used to represent social networks, such as networks of friends on social media.
• Graphs can be used to represent the topology of computer networks, such as the connections between routers and switches.
• Graphs are used to represent the connections between different places in a transportation network, such as roads and airports.
• Neural Networks: Vertices represent neurons and edges represent the synapses between them. Neural networks are used to understand how our brain works and how connections change when we learn. The human brain has about 10^11 neurons and close to 10^15 synapses.
• Compilers: Graphs are used extensively in compilers. They can be used for type inference, for so-called data flow analysis, register allocation, and many other purposes. They are also used in specialized compilers, such as query optimization in database languages.
• Robot planning: Vertices represent states the robot can be in and the edges the possible transitions between the states. Such graph plans are used, for example, in planning paths for autonomous vehicles.

Advantages of Graphs:

• Graphs are a versatile data structure that can be used to represent a wide range of relationships and data structures.
• They can be used to model and solve a wide range of problems, including pathfinding, data clustering, network analysis, and machine learning.
• Graph algorithms are often very efficient and can be used to solve complex problems quickly and effectively.
• Graphs can be used to represent complex data structures in a simple and intuitive way, making them easier to understand and analyze.

Disadvantages of Graphs:

• Graphs can be complex and difficult to understand, especially for people who are not familiar with graph theory or related algorithms.
• Creating and manipulating graphs can be computationally expensive, especially for very large or complex graphs.
• Graph algorithms can be difficult to design and implement correctly, and can be prone to bugs and errors.
• Graphs can be difficult to visualize and analyze, especially for very large or complex graphs, which can make it challenging to extract meaningful insights from the data.

Frequently Asked Questions(FAQs) on Graphs:

1. What is a graph?

A graph is a data structure consisting of a set of vertices (nodes) and a set of edges that connect pairs of vertices.

2. What are the different types of graphs?

Graphs can be classified into various types based on properties such as directionality of edges (directed or undirected), presence of cycles (acyclic or cyclic), and whether multiple edges between the same pair of vertices are allowed (simple or multigraph).

3. What are the applications of graphs?

Graphs have numerous applications in various fields, including social networks, transportation networks, computer networks, recommendation systems, biology, chemistry, and more.

4. What is the difference between a directed graph and an undirected graph?

In an undirected graph, edges have no direction, meaning they represent symmetric relationships between vertices. In a directed graph (or digraph), edges have a direction, indicating a one-way relationship between vertices.

5. What is a weighted graph?

A weighted graph is a graph in which each edge is assigned a numerical weight or cost. These weights can represent distances, costs, or any other quantitative measure associated with the edges.

6. What is the degree of a vertex in a graph?

The degree of a vertex in a graph is the number of edges incident to that vertex. In a directed graph, the indegree of a vertex is the number of incoming edges, and the outdegree is the number of outgoing edges.

7. What is a path in a graph?

A path in a graph is a sequence of vertices connected by edges. The length of a path is the number of edges it contains.

8. What is a cycle in a graph?

A cycle in a graph is a path that starts and ends at the same vertex, traversing a sequence of distinct vertices and edges in between.

9. What are spanning trees and minimum spanning trees?

A spanning tree of a graph is a subgraph that is a tree and includes all the vertices of the original graph. A minimum spanning tree (MST) is a spanning tree with the minimum possible sum of edge weights.

10. What algorithms are commonly used to traverse or search graphs?

Common graph traversal algorithms include depth-first search (DFS) and breadth-first search (BFS). These algorithms are used to explore or visit all vertices in a graph, typically starting from a specified vertex. Other algorithms, such as Dijkstra’s algorithm and Bellman-Ford algorithm, are used for shortest path finding.

Conclusion:

• Graph data structures are a powerful tool for representing and analyzing relationships between objects or entities.
• Graphs can be used to represent the interactions between different objects or entities, and then analyze these interactions to identify patterns, clusters, communities, key players, influencers, bottlenecks and anomalies. 
• In sports data science, graph data structures can be used to analyze and understand the dynamics of team performance and player interactions on the field.
•  They can be used in a variety of fields such as Sports, Social media, transportation, cybersecurity and many more.

More Resources of Graph:

• Recent Articles on Graph
• Practice problems on Graph
• Algorithms on Graphs