Complete Graphs $K_n$

A complete graph is a simple undirected graph in which every pair of distinct vertexes $n$ is connected by a unique edge. This type of graph is denoted by $K_n$.

In other words, a simple graph $G$ with $n>2$ vertexes is called a complete graph if $\forall v\in V(G)$ we have $\delta(v)=n-1$.

properties $\lambda, \kappa$

$$ \lambda(K_n)=\text{min}_{v\in V(G)}\{\delta(v)\}=n-1\\ \kappa(K_n)=\text{Undefined} $$

$r-$Connectivity

A graph is $r$-connected if

• $\kappa(G)\geq r$
• $\delta(G)\geq r$

Harary Graph

Given two numbers $r$ and $n$, the Harary Graph $H_{r, n}$ is an $r$-connected graph with $n$ vertexes and with a minimum number of edges

making a Harary Graph

CASE 1 - $r$ is even

1. Place vertexes $\{0, 1, ..., n-1\}$ on the boundary of a circle
2. Connect each vertex to its $r/2$ nearest left-hand neighbors and to its $r/2$ nearest right hand neighbors

CASE 2 - $r$ is odd and $n$ is even

1. Construct $H_{r-1, n}$ (follow the steps in CASE 1 for the graph $H_{r-1, n}$)
2. Add edges $\langle0, n/2\rangle, \langle 1, 1+n/2\rangle, ...,\langle\frac{n-2}{2}, n-1\rangle$

CASE 3 - $r$ is odd and $n$ is odd

1. Construct $H_{r-1, n}$ (follow the steps in CASE 1 for the graph $H_{r-1, n}$)
2. Add edges $\langle 0, \frac{n-1}{2}\rangle, \langle1, 1+\frac{n-1}{2}\rangle, ..., \langle\frac{n-1}{2}, n-1\rangle$