Connected graphs on $\omega$ that "doesn't change" whenever 2 points are collapsed

Question

For any set $X$, let $[X]^2 = \big\{\{x,y\}:x\neq y\in x\big\}$.

Let $G=(\omega,E)$ be a connected simple, undirected graph, where $E\subseteq [\omega]^2$.. For $m<n\in \omega$ we let $G/_{\{m,n\}}$ be the graph that we get from removing $n$, and $m$ gets all the neighbours from $n$. Formally, let $$G/_{\{m,n\}} = \big(\omega\setminus \{n\},\; (E\cap [\omega\setminus\{n\}]^2)\cup\{\{m, x\}: \{n, x\}\in E\} \land x\neq m\big).$$

We say that $G=(\omega, V)$ has the collapse-property if $G\cong G/_{\{m,n\}}$ for all $m<n\in\omega$.

What is an example of a non-complete, connected graph $G=(\omega,E)$ having the collapse-property?

Answer 1

There are many such graphs:

• The (countable) random graph.

• The "infinite depth-$1$ tree" given by $E=\{\{0,i\}: i>0\}$.

• The "majestic sunflower" consisting of infinitely many $n$-cycles, for each $n$, joined at a common point.