From: Consistency of the Neighbor-Net Algorithm
Case
∑ l = 1 r − 1 d S ( x l , x l + 1 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaaeWaqaaiabdsgaKnaaBaaaleaacqWGtbWuaeqaaOGaeiikaGIaemiEaG3aaSbaaSqaaiabdYgaSbqabaGccqGGSaalcqWG4baEdaWgaaWcbaGaemiBaWMaey4kaSIaeGymaedabeaakiabcMcaPaWcbaGaemiBaWMaeyypa0JaeGymaedabaGaemOCaiNaeyOeI0IaeGymaedaniabggHiLdaaaa@4289@
d S (x1, x r )
∑ l = 1 n d S ( x 1 , x l ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaaeWaqaaiabdsgaKnaaBaaaleaacqWGtbWuaeqaaOGaeiikaGIaemiEaG3aaSbaaSqaaiabigdaXaqabaGccqGGSaalcqWG4baEdaWgaaWcbaGaemiBaWgabeaakiabcMcaPaWcbaGaemiBaWMaeyypa0JaeGymaedabaGaemOBa4ganiabggHiLdaaaa@3E61@
(i)
1
n - j
(ii)
0
(iii)
2
j - i + 1
(iv)
(v)
∑ l = 2 r − 1 ∑ k = 1 n d S ( x l , x k ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaaeWaqaamaaqadabaGaemizaq2aaSbaaSqaaiabdofatbqabaGccqGGOaakcqWG4baEdaWgaaWcbaGaemiBaWgabeaakiabcYcaSiabdIha4naaBaaaleaacqWGRbWAaeqaaOGaeiykaKcaleaacqWGRbWAcqGH9aqpcqaIXaqmaeaacqWGUbGBa0GaeyyeIuoaaSqaaiabdYgaSjabg2da9iabikdaYaqaaiabdkhaYjabgkHiTiabigdaXaqdcqGHris5aaaa@4773@
∑ l = 1 n d S ( x r , x l ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaaeWaqaaiabdsgaKnaaBaaaleaacqWGtbWuaeqaaOGaeiikaGIaemiEaG3aaSbaaSqaaiabdkhaYbqabaGccqGGSaalcqWG4baEdaWgaaWcbaGaemiBaWgabeaakiabcMcaPaWcbaGaemiBaWMaeyypa0JaeGymaedabaGaemOBa4ganiabggHiLdaaaa@3EDE@
(j - 1)(n - j) + (r - j - 1)j
j
(r - 2)(n - j)
(j - i + 1)(n - 2j + 2i + r - 4)
(i - 2)(j - i + 1) + (r - i)(i - 1 + n - j)
i - 1 + n - j
(r - 2)(j - i + 1)