04-25-2014 Enriched with more details
Let define the graphs $(V_\alpha,\in_\alpha)$ in $ZFC$
$V_\alpha$ can be finite too
$\in_\alpha \subseteq V_\alpha \times V_\alpha$ and $\in_\alpha:=\{(a,b):a \in b\}$
note 1: I'm very curious about the links betwen this collection of graphs that I'm describing and the axioms that they satisfies, if I'm not wrong they satisfie $ZFC$ only if $\alpha$ is inaccesible and for $\alpha=\omega$ we have a model of sets without axiomm of infinity. but what we have for smaller ordinals or finite?
$Q1.$ Do these structures have a special name as graphs? And what is the deep link with set theory?
I tried to draw the graphs of the first 4 cases $V_1$, $V_2$, $V_3$ and $V_4$ and I noticed that the nubers of edges of each graph are $$|\in_{\alpha+1}|=\sum_{a \in V_{\alpha+1} }|a| $$
note 2: For finite ordinals the thing is more interesting since it is always the sum of ${^\alpha}2-1$ non-empty sets (where $^{n}a$ is tetration) we "should" have
$$|V_{\alpha+1}\setminus \varnothing|={^\alpha}2-1 \leq|\in_\alpha|$$
even if I don't know if it holds always (I mean for infinite ordinals) but the best I could find was the following...but i'm not sure is correct
$$|\in_{\alpha+2}|=\sum_{i=1 }^{{^\alpha}2}i({^\alpha}2-i+1) $$
$Q2.$ How I can find a closed form for $|\in_{\alpha}| $ and $\alpha<\omega$? What is the pattern for infinite ordinals, what is the upper bound of $|\in_\alpha|$?
I noticed that in the graphs $(V_\alpha,\in_\alpha)$ we can define the cardinality and some other concepts of set theory but limited to $V_\alpha$ and an interesting concept of "inverse cardinality" too.
Elements of a node are the "predecessors" $$elem_{\in_\alpha}(a):=\{b: b\in_\alpha a\}$$
and the cardinality of a node $$|a|_{\in_\alpha}=|elem_{\in_\alpha}(a)|$$
The "owners" of a node are the "successors" $$own_{\in_\alpha}(a):=\{b: a\in_\alpha b\}$$
now my idea ... we can define the notion of "inverse cardinality" with the "owners" function
inverse cardinality of a node $$||a||_{\in_\alpha}=|own_{\in_\alpha}(a)|$$
example "$\{V_1\}$" appears as node in all the graphs $(V_\alpha,\in_\alpha)$ for $\alpha>2$ and while $|\{V_1\}|_{\in_\alpha}=1$ holds for every $\alpha>2$ $||\{V_1\}||_{\in_\alpha}$ doesnt. The value grow:
$||\{V_1\}||_{\in_3}=0$, $||\{V_1\}||_{\in_4}=9$
note 3 we now can see that the number of edges of a graph are exatly the sum of the cardinalities or of the inverse cardinalities of the nodes that graph
$$ |\in_\alpha|=\sum_{a \in V_\alpha}|a|_{\in_\alpha}=\sum_{a \in V_\alpha}||a||_{\in_\alpha} $$
but, as Asaf Karagila pointed out, the value the inverse cardinality of the nodes of start to assume a constant (infinite) value when $\alpha$ gets enough big. In the case of ZFC, in fact, we can build infinite sets that have a set $a$ as element.
$Q3.$ Exist a more general concept of "inverse cardinality" in $ZFC$ for normal sets? Does have a special name/properties? There is a formula for it for graphs with finite nodes ($V_{\alpha<\omega}$) (see the example)?
After editing and adding info in this question I can feel that I did some big mistake somewhere... but I don't know where. I hope that who try to answer will point out my misconceptions too.
Bounty added: reference for standard terminology is asked, and at least one of the question Q2 and Q3