Open Problem Garden - Extremal problem on the number of tree endomorphism - Comments

it is not open (re: Extremal problem on the number of tree endomorphism)

Anonymous — Wed, 25 Nov 2020 09:59:52 +0100

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

the upper bound is proved (re: Extremal problem on the number of tree endomorphism)

Anonymous — Fri, 23 Mar 2012 14:25:47 +0100

the upper bound is proved recently.

counterexample (re: Extremal problem on the number of tree endomorphism)

leshabirukov — Sun, 20 Mar 2011 18:15:41 +0100

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.

Extremal problem on the number of tree endomorphism

shudeshijie — Tue, 01 Mar 2011 13:04:49 +0100

Author(s): Zhicong Lin

Subject: Graph Theory » Extremal G.T.

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.