# Yet another graph characteristic

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

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 $$\sum_{k=0}^n f(k) = n$$.

My questions are:

Have some of these concepts be found useful in graph theory? If so, under which names?

How can they be related to other graph characteristics?

Which functions $$f: \{0,\dots,n\} \rightarrow \{0,\dots,n\}$$ with $$\sum_{k=0}^n f(k) = n$$ can not be cycle number spectra of any graph?