# Extremal problem on the number of tree endomorphism

**Conjecture**An endomorphism of a graph is a mapping on the vertex set of the graph which preserves edges. Among all the vertices' trees, the star with vertices has the most endomorphisms, while the path with vertices has the least endomorphisms.

## Bibliography

[BT] Bela Bollobas and Mykhaylo Tyomkyn, Walks and paths in trees, http://arxiv.org/abs/1002.2768.

* indicates original appearance(s) of problem.

### counterexample

On March 20th, 2011 leshabirukov says:

Asymmetric tree (http://en.wikipedia.org/wiki/Asymmetric_graph, http://upload.wikimedia.org/wikipedia/commons/a/ad/Asymmetric_tree.svg) has single, trivial endomorphism.

Update: Sorry, I have confused endomorphism with automorphism.

## it is not open

This problem is not open. Look at this: https://mathscinet.ams.org/mathscinet/search/publdoc.html?r=1&pg1=MR&s1=3284058&loc=fromrevtext