Concept
Concept
Graph theory is a branch of mathematics that studies the properties and applications of graphs, which are structures made up of nodes (vertices) connected by edges. It is fundamental in computer science, network analysis, and combinatorics for solving problems related to connectivity, flow, and optimization.
Concept
An adjacency matrix is a square matrix used to represent a finite graph, where the element at row i and column j indicates the presence (and sometimes weight) of an edge between vertices i and j. It is a fundamental tool in graph theory, offering a straightforward way to store and manipulate graph data, especially for dense graphs.
Concept
Concept
Concept
A complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. It is characterized by having the maximum possible number of edges for a given number of vertices, making it a fundamental structure in graph theory for studying connectivity and network topology.
Concept
A sparse graph is a type of graph in which the number of edges is significantly lower than the maximum possible number of edges, often making it computationally efficient to store and process. sparse graphs are especially relevant in fields like network theory and computer science, where they model real-world systems with limited connections, such as social networks and transportation grids.
Concept
Graph clustering is the process of grouping vertices in a graph such that vertices within the same group are more densely connected to each other than to those in other groups. It is a crucial technique for simplifying complex networks and uncovering hidden structures in data, applicable in various fields such as social network analysis, biology, and computer science.
Concept
Network analysis is a method used to study the structure and dynamics of complex networks by examining the relationships between interconnected entities. It is widely applied in various fields such as sociology, biology, and computer science to uncover patterns, optimize systems, and predict behaviors.
Concept
Breadth-First Search (BFS) is a fundamental algorithm for traversing or searching tree or graph data structures, exploring all neighbors at the present depth prior to moving on to nodes at the next depth level. It is particularly useful for finding the shortest path in unweighted graphs and is implemented using a queue data structure to keep track of nodes to be explored.
Concept
Depth-First Search (DFS) is an algorithm for traversing or searching tree or graph data structures, prioritizing exploring as far down a branch as possible before backtracking. It is implemented using a stack data structure, either explicitly or through recursion, and is particularly useful for solving problems like pathfinding, cycle detection, and topological sorting in directed graphs.
Concept
Edge addition is a fundamental operation in graph theory, involving the integration of a new edge between two vertices in a graph. This operation can significantly affect the graph’s properties, potentially changing connectivity, influencing pathfinding, or altering the graph's overall structure.
3