### 4.3.4.1. one parameter/one final graph

Proposition 1

$\mathrm{𝚗𝚘}\mathrm{𝚕𝚘𝚘𝚙}:\mathrm{𝐌𝐀𝐗}_\mathrm{𝐍𝐂𝐂}\ne 1$

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

Proposition 2

$\mathrm{𝚊𝚌𝚢𝚌𝚕𝚒𝚌}:\mathrm{𝐌𝐀𝐗}_\mathrm{𝐍𝐒𝐂𝐂}\le 1$

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

Proposition 3

$\mathrm{𝚗𝚘}\mathrm{𝚕𝚘𝚘𝚙}:\mathrm{𝐌𝐀𝐗}_\mathrm{𝐍𝐒𝐂𝐂}\ne 1$

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

Proposition 4

$\mathrm{𝚗𝚘}\mathrm{𝚕𝚘𝚘𝚙}:\mathrm{𝐌𝐈𝐍}_\mathrm{𝐍𝐂𝐂}\ne 1$

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

Proposition 5

$\mathrm{𝚊𝚌𝚢𝚌𝚕𝚒𝚌}:\mathrm{𝐌𝐈𝐍}_\mathrm{𝐍𝐒𝐂𝐂}\le 1$

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

Proposition 6

$\mathrm{𝚗𝚘}\mathrm{𝚕𝚘𝚘𝚙}:\mathrm{𝐌𝐈𝐍}_\mathrm{𝐍𝐒𝐂𝐂}\ne 1$

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

Proposition 7

$\mathrm{𝚘𝚗𝚎}_\mathrm{𝚜𝚞𝚌𝚌}:\mathrm{𝐍𝐀𝐑𝐂}={\mathrm{𝐍𝐕𝐄𝐑𝐓𝐄𝐗}}_{\mathrm{𝙸𝙽𝙸𝚃𝙸𝙰𝙻}}$

Proof 7 By definition of $\mathrm{𝚘𝚗𝚎}_\mathrm{𝚜𝚞𝚌𝚌}$.

Proposition 8

$\mathrm{𝚗𝚘}\mathrm{𝚕𝚘𝚘𝚙}:2·\mathrm{𝐍𝐂𝐂}\le {\mathrm{𝐍𝐕𝐄𝐑𝐓𝐄𝐗}}_{\mathrm{𝙸𝙽𝙸𝚃𝙸𝙰𝙻}}$

Proof 8 By definition of $\mathrm{𝚗𝚘}\mathrm{𝚕𝚘𝚘𝚙}$, each connected component has at least two vertices.

Proposition 9

$\mathrm{𝚌𝚘𝚗𝚜𝚎𝚌𝚞𝚝𝚒𝚟𝚎}_\mathrm{𝚕𝚘𝚘𝚙𝚜}_\mathrm{𝚊𝚛𝚎}_\mathrm{𝚌𝚘𝚗𝚗𝚎𝚌𝚝𝚎𝚍}:2·\mathrm{𝐍𝐂𝐂}\le {\mathrm{𝐍𝐕𝐄𝐑𝐓𝐄𝐗}}_{\mathrm{𝙸𝙽𝙸𝚃𝙸𝙰𝙻}}+1$

Proof 9 By definition of $\mathrm{𝚌𝚘𝚗𝚜𝚎𝚌𝚞𝚝𝚒𝚟𝚎}_\mathrm{𝚕𝚘𝚘𝚙𝚜}_\mathrm{𝚊𝚛𝚎}_\mathrm{𝚌𝚘𝚗𝚗𝚎𝚌𝚝𝚎𝚍}$.

Proposition 10

$\mathrm{𝚗𝚘}\mathrm{𝚕𝚘𝚘𝚙}:2·\mathrm{𝐍𝐒𝐂𝐂}\le {\mathrm{𝐍𝐕𝐄𝐑𝐓𝐄𝐗}}_{\mathrm{𝙸𝙽𝙸𝚃𝙸𝙰𝙻}}$

Proof 10 By definition of $\mathrm{𝚗𝚘}\mathrm{𝚕𝚘𝚘𝚙}$, each strongly connected component has at least two vertices.

Proposition 11

$\mathrm{𝚜𝚢𝚖𝚖𝚎𝚝𝚛𝚒𝚌}:\mathrm{𝐍𝐒𝐈𝐍𝐊}=0$

Proof 11 Since we do not have any isolated vertex.

Proposition 12

$\mathrm{𝚜𝚢𝚖𝚖𝚎𝚝𝚛𝚒𝚌}:\mathrm{𝐍𝐒𝐎𝐔𝐑𝐂𝐄}=0$

Proof 12 Since we do not have any isolated vertex.

Proposition 13

$\mathrm{𝚘𝚗𝚎}_\mathrm{𝚜𝚞𝚌𝚌}:\mathrm{𝐍𝐕𝐄𝐑𝐓𝐄𝐗}={\mathrm{𝐍𝐕𝐄𝐑𝐓𝐄𝐗}}_{\mathrm{𝙸𝙽𝙸𝚃𝙸𝙰𝙻}}$

Proof 13 By definition of $\mathrm{𝚘𝚗𝚎}_\mathrm{𝚜𝚞𝚌𝚌}$.