Properties of Some Random Graphs Working in a problem the following family of graphs appears naturally. Consider the set $A_{n}=\{1,2,3,\ldots,n\}$ and let $\mathcal{C_{n}}$ be the set of all permutations of $A_{n}$ of order $n$ (cycles of order $n$). Let $\sigma_{1},\sigma_{2},\ldots,\sigma_{d}$ be random elements chosen uniformly and without repetition from $\mathcal{C}_{n}$.
Now we construct the random graph $\mathcal{G}$ where the node set is $A_{n}$ and there is an edge between each node $i$ and $\sigma_{p}(i)$ for every $p\in\{1,2,\ldots,d\}$. It's clear that every node in the graph $\mathcal{G}$ has degree at most $2d$ (we ignore multiple edges and loops).

My questions are:
  
  
*
  
*Did anybody studied these graphs before?
  
*Is it known what is the asymptotic diameter of $\mathcal{G}$ for fixed
  $d$ as $n$ increases with high
  probability?
  
*Estimates on the Cheeger constant? Laplacian?
  

 A: For those interested in further reading about the contiguity of the above mentioned models for random regular graphs and generalization of the above:
Catherine Greenhill, Svante Janson, Jeong Han Kim and Nicholas C. Wormald, Permutation pseudographs and contiguity, Combinatorics, Probability and Computing 11 (2002), 273 - 298.
Link: http://web.maths.unsw.edu.au/~csg/papers/gjkw-revised.pdf
A: Yes, this model has been studied. You should look at Chapter 9 of Janson, Luczak and Rucinski's Random Graphs book, and in particular at Corollary 9.44. This corollary is in fact a rather well-known theorem, which I'll now explain.
Let $H_n(d)$ be the distribution you describe (Edit: more accurately, $H_n(d)$ is the distribution of the union of $d$ independent and uniformly random cycles, conditioned on the result being a simple graph), and let $G_n(2d)$ be the distribution of a uniformly random $2d$-regular (all nodes having degree exactly $2d$) simple graph. Then Corollary 9.44 states that for any fixed $d$, $H_n(d)$ and $G_n(2d)$ are contiguous, which means that for any graph property $A$, 
$$
\mathbb{P}(H_n(d) \in A) \to 1~\mbox{as}~n\to\infty
$$
if and only if
$$\mathbb{P}(G_n(2d) \in A) \to 1~\mbox{as}~n\to\infty.$$
In other words, if you are only interested in studying whether things hold asymptotically almost surely, these two models are equivalent. 
In particular, its isoperimetric constant is $(1/2+o(1)) d$, its diameter is $(1+o(1)) \log_{d-1} (n)$, and all eigenvalues except for the largest are $\sqrt{2(d-1)}+o(1)$. 
