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In computer science, a graph is an abstract data type that is meant to implement the undirected graph and directed graph concepts from the field of graph theory within mathematics. A graph data structure consists of a finite (and possibly mutable) set of vertices (also called nodes or points ), together with a set of unordered pairs of these vertices for an undirected graph or a set of ordered ...
- Overview
- The Basics of Graph Theory
- Types of Graphs
- Conclusions
In this tutorial, we’ll discuss some of the most important data structures in computer science – graphs. We’ll first study the basics of graph theory, in order to familiarize ourselves with its conceptual foundation. We’ll then study the types of graphs that we can find in our machine learning applications. At the end of this tutorial, we’ll know w...
2.1. The Definition of a Graph
A graph is a structurethat comprises a set of vertices and a set of edges. So in order to have a graph we need to define the elements of two sets: vertices and edges. The vertices are the elementary units that a graph must have, in order for it to exist. It’s customary to impose on graphs the condition that they must have at least one vertex, but there’s no real theoretical reason why this is the case. Vertices are mathematical abstractions corresponding to objects associated with one another...
2.2. General Properties of Vertices
We’re now going to focus in more detail about what characteristics vertices and edges possess. Let’s start with the vertices first. As stated before, graphs need vertices but don’t necessarily require edges. In fact, it’s perfectly possible to have graphs composed entirely by vertices. Vertices that aren’t connected to any others, such as those of the empty graphs, are called isolated: We also say that isolated vertices have a degree equal to zero. Degree, in this context, indicates the numbe...
2.3. Labels of Vertices
Vertices can also have values associated with them. These values can take any format and there are no specific restrictions for them. A vertex with an associated value is called a labeled vertex, while a vertex with no associated value is called unlabeled: In general, we can distinguish any two unlabeled vertices exclusively on the basis of their paired vertices. The comparison between labeled vertices requires us instead to study both the pairs of vertices and the values assigned to them: On...
3.1. The Empty Graph
We mentioned before that graphs exist only if their set of vertices is not null. Their set of edges, however, may as well be empty. If this is the case, we say that the graph is empty: An empty graph has always size .
3.2. The Directed Graph
As anticipated above, a directed graph is a graph that possesses at least one edge between two vertices and which doesn’t have a corresponding edge connecting the same vertices in the opposite direction. Directed graphs have the characteristic that they model real-world relationships well for which we can’t freely interchange the subject and the object. As a general rule, if we aren’t sure whether a graph should be directed or undirected, then the graph is directed: We can only traverse direc...
3.3. The Undirected Graph
Undirected graphs are graphs for which the existence of any edge between the vertices implies the presence of a corresponding edge : Undirected graphs allow their traversal between any two vertices connected by an edge. The same isn’t necessarily true for directed graphs.
In this tutorial, we studied the conceptual bases of graph theory. We also familiarized ourselves with the definitions of graphs, vertices, edges, and paths. We’ve also studied the types of graphs that we can encounter and what are their predictable characteristics in terms of vertices, edges, and paths.
- Gabriele De Luca
Aug 5, 2024 · Disconnected Graph: A disconnected graph has isolated components that are not connected to each other. These components are separate subgraphs. 10. Tree. A Tree is a connected graph with no cycles. It is a fundamental data structure in computer science, commonly used in algorithms like binary search trees and heap data structures.
Aug 5, 2024 · Applications of Graph Theory: In mathematics and computer science, a graph is a mathematical structure that consists of two main components: vertices (or nodes) and edges. The study of these graphs in various contexts is called graph theory. There are various applications of graph theory in real life such as in computer graphics and networks, biolo
May 26, 2023 · A directed graph. G raphs serve as a fundamental data structure in computer science, enabling the representation and analysis of complex relationships between entities. Graph theory provides a ...
In mathematics and computer science, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of vertices (also called nodes or points ) which are connected by edges (also called arcs , links or lines ).
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Aug 21, 2024 · In computer science, graph theory underpins many algorithms and data structures used to represent networks such as the internet, social networks, and communication systems. It also provides tools for solving problems related to network connectivity, pathfinding, and optimization, and is foundational for understanding various mathematical structures and concepts such as trees, cycles, and ...
