10.1 Graph Terminology and Properties

A graph is a mathematical structure used to represent relationships between pairs of objects, consisting of vertices (or nodes) connected by edges. In data structures, graphs are essential for modeling various real-world scenarios such as networks, social connections, and pathways. They can be directed or undirected, weighted or unweighted, and provide a versatile way to analyze complex relationships in data.

Vertex: A vertex (or node) is a fundamental unit of a graph, representing an object or point in the structure that can be connected to other vertices by edges.

Edge: An edge is a connection between two vertices in a graph, which can represent various types of relationships depending on the context of the graph.

Adjacency List: An adjacency list is a common way to represent a graph using an array of lists, where each list corresponds to a vertex and contains all the adjacent vertices connected by edges.

A vertex is a fundamental part of a graph, representing a distinct point where edges meet. It serves as a key component in various graph representations and plays a crucial role in the properties and operations related to graphs, including traversal algorithms and data structure implementations.

edge: An edge is a connection between two vertices in a graph, which can be directed or undirected, indicating the relationship or pathway between them.

degree: The degree of a vertex is the number of edges connected to it, which can provide insights into its connectivity and role within the graph.

adjacency list: An adjacency list is a common graph representation method where each vertex maintains a list of its adjacent vertices, effectively showing all connections from that vertex.

An edge is a fundamental component in graph theory, representing a connection or relationship between two vertices (or nodes) in a graph. In the context of various data structures, edges can signify relationships in hierarchical structures like trees, or connections in more complex networks, playing a critical role in understanding and navigating these structures.

Vertex: A vertex (or node) is an individual element in a graph or tree structure, which can represent objects or points of interest.

Adjacency List: An adjacency list is a way of representing a graph where each vertex stores a list of adjacent vertices connected by edges.

Weighted Edge: A weighted edge is an edge that has an associated numerical value (weight) representing the cost, distance, or capacity of the connection between two vertices.

An undirected graph is a set of objects called vertices connected by edges, where the edges have no direction. This means that if there is an edge connecting vertex A to vertex B, you can traverse from A to B and from B to A freely, highlighting the bidirectional nature of connections. In these graphs, relationships are symmetrical, making them suitable for representing mutual relationships like friendships or road networks.

Directed Graph: A graph where the edges have a direction, indicating a one-way relationship between vertices.

Weighted Graph: A graph in which each edge has an associated weight or cost, often used to represent distances or values.

Adjacency List: A data structure that represents a graph by storing a list of all adjacent vertices for each vertex.

A directed graph, or digraph, is a set of vertices connected by edges, where each edge has a direction associated with it. This means that the edges are ordered pairs, indicating a one-way relationship from one vertex to another. Directed graphs are essential in representing structures where relationships are not mutual, such as web pages linking to each other or tasks that depend on others.

Vertex: A vertex is a fundamental part of a graph that represents an entity, such as a node in a network.

Edge: An edge is a connection between two vertices in a graph, which can be directed or undirected based on whether the relationship has a direction.

Adjacency List: An adjacency list is a way of representing a graph where each vertex has a list of other vertices to which it is connected by edges.

In-degree refers to the number of incoming edges directed toward a particular vertex in a directed graph. It provides insight into the connectivity and structure of the graph by indicating how many other vertices are connected to the specific vertex. This measure is crucial for understanding flow, paths, and relationships within the graph, and it plays a key role in algorithms that analyze directed networks.

out-degree: Out-degree is the number of outgoing edges from a specific vertex in a directed graph, showing how many vertices the vertex points to.

directed graph: A directed graph is a set of vertices connected by edges where each edge has a direction, indicating a one-way relationship between vertices.

vertex: A vertex (or node) is a fundamental unit in a graph that represents an entity, such as an object or location, and is connected to other vertices via edges.

Out-degree refers to the number of edges that originate from a given vertex in a directed graph. This concept is essential for understanding the structure and behavior of directed graphs, as it helps determine how many connections a particular vertex has leading away from it. Out-degree plays a crucial role in analyzing graph properties such as connectivity, flow, and traversal algorithms.

in-degree: In-degree is the number of edges directed toward a given vertex in a directed graph.

vertex: A vertex is a fundamental unit of a graph, representing an entity or node in the context of graph theory.

edge: An edge is a connection between two vertices in a graph, representing a relationship or link between them.

A complete graph is a type of graph in which every pair of distinct vertices is connected by a unique edge. This means that in a complete graph, there are no missing edges, making it fully connected. The number of edges in a complete graph can be calculated using the formula $$E = \frac{n(n-1)}{2}$$, where 'n' is the number of vertices. Complete graphs are important because they represent the most connected structure possible among a set of points.

Vertex: A point in a graph where edges meet, representing an entity or a position.

Edge: A connection between two vertices in a graph, representing the relationship or interaction between them.

Graph Theory: The branch of mathematics that studies graphs and their properties, including how they can represent relationships in various structures.