The following random construction is simple enough that I am guessing it must have been studied. Fix $d \ge 3$, and let $n > d$. For each of the $n$ vertices, pick exactly $d$ other vertices to connect it to, uniformly over all ${n-1 \choose d}$ possible choices, and making this choice independently over all $n$ vertices. Some edges may get put in twice, and that's fine, but we will consider the final graph to not have multiple edges.



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Motivation: Erdos-Renyi random graphs $G(n,p)$ must have average vertex degree greater than, before they become connected and have nice expansion properties, etc. But this is because they still have isolated vertices. Here we don't have the problem of isolated vertices. (I think I can show that these graphs are a.a.s. connected.) I am aware of the large literature on random $d$-regular graphs, but I thought this might be an alternative model if one does not care about regularity.


I am especially wondering about spectral properties of these graphs.



Has the following kind of (minimum degree $d$) random graph been studied?
Yes, this model has been studied. For some early results, see


Fenner, T. I.; Frieze, A. M. On the connectivity of random $m$-orientable graphs and digraphs. Combinatorica 2 (1982), no. 4, 347–359.


So there it is called "random $m$-orientable graph"


