Complete Graph is Regular
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Theorem
Let $K_p$ be the complete graph of order $p$.
Then $K_p$ is $p-1$-regular.
Proof
By definition of complete graph, $K_p$ has $p$ vertices.
Also by definition of complete graph, each vertex of $K_p$ is adjacent to all the other $p - 1$ vertices of $K_p$.
As $K_p$ is a simple graph, there can be only one edge joining any pair of vertices of $K_p$.
So each vertex of $K_p$ has $p - 1$ edges to which it is incident.
So, by definition, $K_p$ is $p-1$-regular.
$\blacksquare$
Examples
Complete Graph $K_5$
The complete graph $K_5$ of order $5$ is $4$-regular.
Sources
- 1977: Gary Chartrand: Introductory Graph Theory ... (previous) ... (next): Chapter $2$: Elementary Concepts of Graph Theory: $\S 2.1$: The Degree of a Vertex