## Complete graphs

The complete graph $K_n$, $n\geq1$ is the graph with $n$ vertices, such that every pair of vertices is adjacent.

## Paths

The path $P_n$, $n\geq2$ is the graph with $V(P_n)=\{1,2,\ldots,n\}$ and

$\displaystyle E(P_n)=\{\{1,2\},\{2,3\},\ldots,\{n-1,n\}\}$.

## Cycles

The cycle $C_n$, $n\geq3$ is the graph with $V(C_n)=\{1,2,\ldots,n\}$ and

$\displaystyle E(C_n)=E(P_n)\cup\{\{1,n\}\}$.

## Complete bipartite graphs

The complete bipartite graph $K_{m,n}$, $m\geq1$, $n\geq1$ is the graph where:

$\displaystyle V(K_{m,n})=\{(1,i)\mid i=1,2,\ldots,m\}\cup\{(2,j)\mid i=1,2,\ldots,n\}$

and $(i,r)\sim (j,s)$ if and only if $i\ne j$.