I wonder if the following graph-theoretical concepts have been considered before, and if so, under which name. 


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Consider a directed graph $G$ with $n$ nodes.

Let the **cycle number** $\gamma(\nu)$ be the length of the shortest directed cycle from node $\nu$ to itself. $\gamma(\nu) = 1$ when $\nu$ is connected to itself. Let $\gamma(\nu) = 0$ when there is no cycle from $\nu$ to itself.

Let the **mean cycle number** $\overline{\gamma}(G)$ be $\frac{1}{n}\sum_{i=1}^n \gamma(\nu_i)$.

Let the **shortcutness** $\sigma(e)$ of edge $e$ be the number

$$\sigma(e) = 1 - \frac{\overline{\gamma}(G\setminus\{e\})}{\overline{\gamma}(G)}$$

When $\sigma(e)=0$, i.e. $\overline{\gamma}(G\setminus\{e\}) = \overline{\gamma}(G)$, then edge $e$ doesn't act as a shortcut. 

When $\sigma(e)=1$, i.e. $\overline{\gamma}(G\setminus\{e\}) = 0$, then edge $e$ is contained in every cycle and thus is an ubiquitous shortcut.

Let the **cycle number spectrum** be the function $f: \{0,\dots,n\} \rightarrow \{0,\dots,n\}$ with $f(k)$ being the number of nodes $\nu$ with $\gamma(\nu) = k$. We have $\sum_{k=0}^n f(k) = n$.

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My questions are:

> Have some of these concepts be found useful in graph theory? If so,
> under which names?
> 
> Can the cycle number spectrum be related to other graph
> characteristics? 
> 
> Which functions $f: \{0,\dots,n\} \rightarrow \{0,\dots,n\}$ can not
> be cycle number spectra of any graph?




  [1]: https://mathoverflow.net/questions/273268/another-graph-characteristic