induced subgraphs

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\begin{document}

For every connected undirected graph $G$, there exists a function $s_G$ with two parameters $n$ and $m$ defined like so:

\bigskip

$s_G(n, m)$ = Maximum number of distinct, connected subgraphs of $G$ of order $n$, in which each vertex of $G$ is used in at most $m$ of these subgraphs.

For example, in this graph:

\includegraphics[width=0.5\textwidth]{Download__2_.png}

$s(1, 1) = |V| = 5$

$s(2, \infty) = |E| = 4$

$s(2, 1) = 1$ (in fact for all diameter-2 graphs, $s(2, m) = m$ up to $|E|$)

Another useful property:

$s(|V| - 1, \infty) = $ number of non-articulation points in the graph.

\bigskip

My question is, does this $s_G$ function unique determine graph $G$?

In other words, are two graphs $G$ and $G'$ isomorphic if and only if they have the same function?

And if so, if you restrict the second parameter $m$ to the two values of $\{1, \infty\}$ does this new function also do the same?

\end{document}