A graph having an edge from each vertex to every other vertex is called a ___________

A graph having an edge from each vertex to every other vertex is called a complete graph.

In a complete graph, every pair of distinct vertices is connected by a unique edge. The notation for a complete graph with $n$ vertices is denoted as $K_n$. The characteristics of a complete graph include:

1. Number of Edges: The number of edges in a complete graph can be calculated using the formula: $E = \frac{n(n-1)}{2}$ where $E$ is the number of edges, and $n$ is the number of vertices.

2. Fully Connected: Each vertex has a direct connection to every other vertex, making it a highly connected graph.

3. Properties: A complete graph is always simple (no loops or multiple edges) and possesses $n$ vertices and $E$ edges as calculated above.

Complete graphs are fundamental in graph theory and have applications in various fields including network design, social network analysis, and combinatorial optimization.