### 4.3.4.1. one parameter/one final graph

Proposition 1

Proof 1 Since we do not have any loop, a non -empty connected component has at least two vertices.

Proposition 2

$\mathrm{\pi \pi \pi ’\pi \pi \pi \pi }:\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi }\beta €1$

Proof 2 Since we do not have any circuit, a non -empty strongly connected component consists of one single vertex.

Proposition 3

Proof 3 Since we do not have any loop, a non -empty strongly connected component has at least two vertices.

Proposition 4

Proof 4 Since we do not have any loop, a non -empty connected component has at least two vertices.

Proposition 5

$\mathrm{\pi \pi \pi ’\pi \pi \pi \pi }:\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi }\beta €1$

Proof 5 Since we do not have any circuit, a non -empty strongly connected component consists of one single vertex.

Proposition 6

Proof 6 Since we do not have any loop, a non -empty strongly connected component has at least two vertices.

Proposition 7

$\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi }:\mathrm{\pi \pi \pi \pi }={\mathrm{\pi \pi \pi \pi \pi \pi \pi }}_{\mathrm{\pi Έ\pi ½\pi Έ\pi \pi Έ\pi °\pi »}}$

Proof 7 By definition of $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi }$.

Proposition 8

$\mathrm{\pi \pi }\mathrm{\pi \pi \pi \pi }:2Β·\mathrm{\pi \pi \pi }\beta €{\mathrm{\pi \pi \pi \pi \pi \pi \pi }}_{\mathrm{\pi Έ\pi ½\pi Έ\pi \pi Έ\pi °\pi »}}$

Proof 8 By definition of $\mathrm{\pi \pi }\mathrm{\pi \pi \pi \pi }$, each connected component has at least two vertices.

Proposition 9

$\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }:2Β·\mathrm{\pi \pi \pi }\beta €{\mathrm{\pi \pi \pi \pi \pi \pi \pi }}_{\mathrm{\pi Έ\pi ½\pi Έ\pi \pi Έ\pi °\pi »}}+1$

Proof 9 By definition of $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }$.

Proposition 10

$\mathrm{\pi \pi }\mathrm{\pi \pi \pi \pi }:2Β·\mathrm{\pi \pi \pi \pi }\beta €{\mathrm{\pi \pi \pi \pi \pi \pi \pi }}_{\mathrm{\pi Έ\pi ½\pi Έ\pi \pi Έ\pi °\pi »}}$

Proof 10 By definition of $\mathrm{\pi \pi }\mathrm{\pi \pi \pi \pi }$, each strongly connected component has at least two vertices.

Proposition 11

$\mathrm{\pi \pi ’\pi \pi \pi \pi \pi \pi \pi }:\mathrm{\pi \pi \pi \pi \pi }=0$

Proof 11 Since we do not have any isolated vertex.

Proposition 12

$\mathrm{\pi \pi ’\pi \pi \pi \pi \pi \pi \pi }:\mathrm{\pi \pi \pi \pi \pi \pi \pi }=0$

Proof 12 Since we do not have any isolated vertex.

Proposition 13

$\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi }:\mathrm{\pi \pi \pi \pi \pi \pi \pi }={\mathrm{\pi \pi \pi \pi \pi \pi \pi }}_{\mathrm{\pi Έ\pi ½\pi Έ\pi \pi Έ\pi °\pi »}}$

Proof 13 By definition of $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi }$.