What is graph automorphism and graph homomorphism I know what is graph isomorphism these are isomorpic
But what is graph automorhpism and  graph homomorphism in a simple way, can you post an example of them?
 A: Homomorphisms are the most general structure preserving maps between graphs. You can map any vertex to any other as long as everything which is connected before is also connected afterwards.
For example take a bipartite graph $G=(A\cup B,E)$. We can map this graph into the single-edge-graph $G'=(\{a,b\},\{\{a,b\}\})$ with only two vertices $a$ and $b$ connected by an edge $\{a,b\}$. A possible homomorphism is
$$\phi(v)=\begin{cases}a&\text{for $v\in A$}\\b&\text{for $v\in B$}\end{cases}.$$
Isomorphisms are special homomorphisms which are invertible and for which the inverse is a homomorphism too. The difference to general homomorphisms is that you now must pay additional attention that anything which is not connected before must be not connected afterwards. Something which is not necessary for general homomorphisms.
Automorphisms are special isomorphsism which map from a graph to the same graph. For example, an automorhism of the cycle $C_n:=\{\Bbb Z_n,\{\{i,i+1\}\mid i\in\Bbb Z_n\}\}$ has the automorphism $\phi(i)=i+1$. It can be interpreted as a rotation of the cycle graph.
