4.3.4.1. one parameter/one final graph

Proposition 1

πš—πš˜ πš•πš˜πš˜πš™:πŒπ€π—_𝐍𝐂𝐂≠1

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

Proposition 2

πšŠπšŒπš’πšŒπš•πš’πšŒ:πŒπ€π—_𝐍𝐒𝐂𝐂≀1

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

Proposition 3

πš—πš˜ πš•πš˜πš˜πš™:πŒπ€π—_𝐍𝐒𝐂𝐂≠1

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

Proposition 4

πš—πš˜ πš•πš˜πš˜πš™:𝐌𝐈𝐍_𝐍𝐂𝐂≠1

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

Proposition 5

πšŠπšŒπš’πšŒπš•πš’πšŒ:𝐌𝐈𝐍_𝐍𝐒𝐂𝐂≀1

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

Proposition 6

πš—πš˜ πš•πš˜πš˜πš™:𝐌𝐈𝐍_𝐍𝐒𝐂𝐂≠1

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

Proposition 7

πš˜πš—πšŽ_𝚜𝚞𝚌𝚌:𝐍𝐀𝐑𝐂=𝐍𝐕𝐄𝐑𝐓𝐄𝐗 π™Έπ™½π™Έπšƒπ™Έπ™°π™»

Proof 7 By definition of πš˜πš—πšŽ_𝚜𝚞𝚌𝚌.

Proposition 8

πš—πš˜ πš•πš˜πš˜πš™:2·𝐍𝐂𝐂≀𝐍𝐕𝐄𝐑𝐓𝐄𝐗 π™Έπ™½π™Έπšƒπ™Έπ™°π™»

Proof 8 By definition of πš—πš˜ πš•πš˜πš˜πš™, each connected component has at least two vertices.

Proposition 9

πšŒπš˜πš—πšœπšŽπšŒπšžπšπš’πšŸπšŽ_πš•πš˜πš˜πš™πšœ_πšŠπš›πšŽ_πšŒπš˜πš—πš—πšŽπšŒπšπšŽπš:2·𝐍𝐂𝐂≀𝐍𝐕𝐄𝐑𝐓𝐄𝐗 π™Έπ™½π™Έπšƒπ™Έπ™°π™» +1

Proof 9 By definition of πšŒπš˜πš—πšœπšŽπšŒπšžπšπš’πšŸπšŽ_πš•πš˜πš˜πš™πšœ_πšŠπš›πšŽ_πšŒπš˜πš—πš—πšŽπšŒπšπšŽπš.

Proposition 10

πš—πš˜ πš•πš˜πš˜πš™:2·𝐍𝐒𝐂𝐂≀𝐍𝐕𝐄𝐑𝐓𝐄𝐗 π™Έπ™½π™Έπšƒπ™Έπ™°π™»

Proof 10 By definition of πš—πš˜ πš•πš˜πš˜πš™, each strongly connected component has at least two vertices.

Proposition 11

πšœπš’πš–πš–πšŽπšπš›πš’πšŒ:ππ’πˆππŠ=0

Proof 11 Since we do not have any isolated vertex.

Proposition 12

πšœπš’πš–πš–πšŽπšπš›πš’πšŒ:ππ’πŽπ”π‘π‚π„=0

Proof 12 Since we do not have any isolated vertex.

Proposition 13

πš˜πš—πšŽ_𝚜𝚞𝚌𝚌:𝐍𝐕𝐄𝐑𝐓𝐄𝐗=𝐍𝐕𝐄𝐑𝐓𝐄𝐗 π™Έπ™½π™Έπšƒπ™Έπ™°π™»

Proof 13 By definition of πš˜πš—πšŽ_𝚜𝚞𝚌𝚌.