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Graphs, in the context of mathematics and computer science, are structures comprising a set of objects, known as vertices or nodes, and a set of pairs of these objects, known as edges or links. These pairs define the relationships between the vertices. Graphs are used to model a wide range of phenomena, from social networks to computer networks, and are a fundamental concept in discrete mathematics and theoretical computer science.

A graph can be represented in several ways, including adjacency matrices and adjacency lists. An adjacency matrix is a square matrix used to represent a finite graph. The elements of the matrix indicate whether pairs of vertices are adjacent or not in the graph. An adjacency list is an array of lists. The ith list denotes the set of neighbors of vertex i in the graph.

Graphs can be classified into several types based on their properties. An undirected graph is a graph in which edges have no orientation. The edge (u, v) is identical to the edge (v, u). A directed graph, also called a digraph, is a graph in which edges have a direction associated with them. The edge (u, v) is not necessarily the same as the edge (v, u). A weighted graph is a graph in which a number (the weight) is assigned to each edge. This weight can represent various quantities, such as distance, cost, or capacity.

Graphs are also used in various algorithms, such as shortest path algorithms (e.g., Dijkstra's algorithm, Bellman-Ford algorithm) and network flow algorithms (e.g., Ford-Fulkerson algorithm). These algorithms are used to solve problems in fields such as operations research, computer science, and engineering.