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I asked some questions about finite Cayley graphs with special type of subgraphs which has been answered by Dear Prof. Godsil. It can be seen in the MO page with address:

Cayley graphs and its subgraphs

But there are some new questions which is interesting for me if there are some answers for them.

$1)$ Are there infinite $r$-regular Cayley graph which all its induced $s$-regular subgraphs are Cayley? (non-trivial cases such as $\cup_1^\infty K_2$ or such as @verret introuduced.)

$2)$ Are there infinite $r$-regular graph (which is not Cayley) which all its induced $s$-regular ($s<r$) subgraphs are Cayley?

$3)$ Let $\Gamma_G^S=Cay(G,S)$ be a finite (or infinite but $r$-regular) Cayley graph and $\Gamma'$ be an induced Cayley subgraph of $\Gamma_G^S$. Can we describe the subgraph $\Gamma'$ by some quotient of $G$ or by some its subgroups?

$4)$ Find all groups $G$ (finite or infinite), in a such a way that for arbitrary normal subgroup of $G$, say $N$, the Cayley graph $Cay(G,S)$ has an induced subgraph isomorphic to $Cay(\frac{G}{N},\overline{S})$, where $\overline{S}=SN$.

I appreciate any answers and references for these questions.

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    $\begingroup$ Infinite trees yield a somewhat trivial positive answer to 1). $\endgroup$ – verret Jul 13 '15 at 21:41
  • $\begingroup$ Thanks @verret, you are right, but I interest to non-trivial case or some family of non-trivial cases. $\endgroup$ – Shahrooz Janbaz Jul 13 '15 at 22:10

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