Vertex
A vertex is one of the fundamental elements of a graph. Formally, a vertex is an element of the set \(V\).
Intuitively, vertices represent the objects of the problem modeled by the graph, such as cities, users, states, or points.
The set of vertices is usually denoted by \(V = \{v_1, v_2, \dots, v_n\}\), where \(|V| = n\).
Edge
An edge is a directed connection between two vertices. Formally, an edge is an ordered pair \((u, v)\), with \(u, v \in V\).
The edge \((u, v)\) indicates that there is a connection that starts at vertex u and ends at vertex v.
The set of edge is denoted by \(A \subseteq V \times V\), and the number of edges is given by \(|A| = m\).
Important note
In undirected graphs, connections have no direction and are called edges, being represented by unordered pairs of vertices.
In directed graphs, the term arc is used, emphasizing the existence of direction.