There are at least two questions on this topic but the answers are not clear to me and WiKi link didn't make it any clear either.
Could someone please clarify is the degeneracy of a graph $G$ defined by:
$$\max \{\delta(s):s\in S\}$$
where $S$ is the set of all sub-graphs of $G$ including $G$ and empty set $\varnothing$, and $\delta(s)$ is the minimum degree of the graph $s$ over all its nodes?
By this definition degeneracy of a ring (circle) graph is $2$ and of star graph is $1$.
Yes, that definition is correct. Note that you get the same value if the maximum is taken only over the induced subgraphs of $G$.
An equivalent formulation: the degeneracy of $G$ is the least $k$ such that the vertices of $G$ can be arranged in a sequence so that each vertex is adjacent to at most $k$ of the vertices that follow it in the sequence. [Let $v_1$ be a vertex of minimum degree in $G$, let $v_2$ be a vertex of minimum degree in $G-v_1$, and so on.]
Also equivalent: the degeneracy of $G$ is the least $k$ such that the vertices of $G$ can be arranged in a sequence so that each vertex is adjacent to at most $k$ of the vertices that precede it in the sequence. [Reverse the previous sequence.]
Clearly, if the degeneracy of $G$ is $k$, then the chromatic number of $G$ is at most $k+1$. Thus the chromatic number of a cycle is at most $3$, the chromatic number of a star is at most $2$.