1. **Problem Statement:** Define a graph in the context of discrete mathematics and graph theory, and provide several numerical examples. 2. **Definition:** A graph $G$ is a pair $G = (V, E)$ where: - $V$ is a set of vertices (or nodes). - $E$ is a set of edges, which are unordered pairs of vertices (for undirected graphs) or ordered pairs (for directed graphs). 3. **Explanation:** Each edge connects two vertices. Graphs can be used to model relationships or connections between objects. 4. **Examples:** - Example 1: $V = \{1, 2, 3\}$, $E = \{\{1, 2\}, \{2, 3\}\}$ represents a graph with 3 vertices and 2 edges. - Example 2: $V = \{4, 5, 6, 7\}$, $E = \{\{4, 5\}, \{5, 6\}, \{6, 7\}, \{7, 4\}\}$ represents a cycle graph with 4 vertices. - Example 3: $V = \{10, 20\}$, $E = \{\{10, 20\}\}$ represents a simple graph with 2 vertices connected by one edge. 5. **Summary:** A graph is a collection of points (vertices) connected by lines (edges). The examples show how vertices and edges are represented numerically.