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❆❧❣❡❜r❛✐❝✱ ❝♦♠❜✐♥❛t♦r✐❛❧ ❛♥❞ str✉❝t✉r❛❧ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❙t❛r ❣r❛♣❤s ❊❧❡♥❛ ❑♦♥st❛♥t✐♥♦✈❛ ❙♦❜♦❧❡✈ ■♥st✐t✉t❡ ♦❢ ▼❛t❤❡♠❛t✐❝s✱ ◆♦✈♦s✐❜✐rs❦✱ ❘✉ss✐❛ ◆♦✈♦s✐❜✐rs❦ ❙t❛t❡ ❯♥✐✈❡rs✐t②✱ ◆♦✈♦s✐❜✐rs❦✱ ❘✉ss✐❛ ❈♦♠✐♥❛t♦r✐❝s ❙❡♠✐♥❛r ❙❏❚❯ ✾ ❆♣r✐❧ ✷✵✶✽✱ ❙❤❛♥❣❤❛✐

Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❙t❛r ❣r❛♣❤ ❝♦♥♥❡❝t❡❞ ❜✐♣❛rt✐t❡ ✕r❡❣✉❧❛r ❣r❛♣❤ ♦❢ ♦r❞❡r ❛♥❞ ❞✐❛♠❡t❡r ✭❙✳ ❇✳ ❆❦❡rs✱ ❇✳ ❑r✐s❤♥❛♠✉rt❤②✱ ✮ ✈❡rt❡①✲tr❛♥s✐t✐✈❡ ❛♥❞ ❡❞❣❡✲tr❛♥s✐t✐✈❡ ❝♦♥t❛✐♥s ❤❛♠✐❧t♦♥✐❛♥ ❝②❝❧❡s ✭❱✳ ❑♦♠♣❡❧✬♠❛❦❤❡r✱ ❱✳ ▲✐s❦♦✈❡ts✱ ✱ P✳ ❙❧❛t❡r✱ ✮ ✐t ❞♦❡s ❝♦♥t❛✐♥ ❡✈❡♥ ✕❝②❝❧❡s ✇❤❡r❡ ❤❛s ❤✐❡r❛r❝❤✐❝❛❧ str✉❝t✉r❡ ❤❛s ✐♥t❡❣r❛❧ s♣❡❝tr✉♠ ❉❡✜♥✐t✐♦♥ ❚❤❡ ❙t❛r ❣r❛♣❤ S n = Cay ( Sym n , T ) , n � 2 ✐s t❤❡ ❈❛②❧❡② ❣r❛♣❤ ♦♥ t❤❡ s②♠♠❡tr✐❝ ❣r♦✉♣ Sym n ♦❢ ♣❡r♠✉t❛t✐♦♥s π = [ π 1 π 2 ...π i ...π n ] ✇✐t❤ t❤❡ ❣❡♥❡r❛t✐♥❣ s❡t T ♦❢ ❛❧❧ tr❛♥s♣♦s✐t✐♦♥s t i = (1 i ) s✇❛♣♣✐♥❣ t❤❡ 1 st ❛♥❞ i t❤ ❡❧❡♠❡♥ts ♦❢ ❛ ♣❡r♠✉t❛t✐♦♥ π ✳ ❊❧❡♥❛ ❑♦♥st❛♥t✐♥♦✈❛ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❙t❛r ❣r❛♣❤s ❙❏❚❯✲✷✵✶✽ ✷ ✴ ✸✺

❉❡✜♥✐t✐♦♥ ❚❤❡ ❙t❛r ❣r❛♣❤ S n = Cay ( Sym n , T ) , n � 2 ✐s t❤❡ ❈❛②❧❡② ❣r❛♣❤ ♦♥ t❤❡ s②♠♠❡tr✐❝ ❣r♦✉♣ Sym n ♦❢ ♣❡r♠✉t❛t✐♦♥s π = [ π 1 π 2 ...π i ...π n ] ✇✐t❤ t❤❡ ❣❡♥❡r❛t✐♥❣ s❡t T ♦❢ ❛❧❧ tr❛♥s♣♦s✐t✐♦♥s t i = (1 i ) s✇❛♣♣✐♥❣ t❤❡ 1 st ❛♥❞ i t❤ ❡❧❡♠❡♥ts ♦❢ ❛ ♣❡r♠✉t❛t✐♦♥ π ✳ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❙t❛r ❣r❛♣❤ ❝♦♥♥❡❝t❡❞ ❜✐♣❛rt✐t❡ ( n − 1) ✕r❡❣✉❧❛r ❣r❛♣❤ ♦❢ ♦r❞❡r n ! ❛♥❞ ❞✐❛♠❡t❡r diam ( S n ) = ⌊ 3( n − 1) ⌋ ✭❙✳ ❇✳ ❆❦❡rs✱ ❇✳ ❑r✐s❤♥❛♠✉rt❤②✱ 1989 ✮ 2 ✈❡rt❡①✲tr❛♥s✐t✐✈❡ ❛♥❞ ❡❞❣❡✲tr❛♥s✐t✐✈❡ ❝♦♥t❛✐♥s ❤❛♠✐❧t♦♥✐❛♥ ❝②❝❧❡s ✭❱✳ ❑♦♠♣❡❧✬♠❛❦❤❡r✱ ❱✳ ▲✐s❦♦✈❡ts✱ 1975 ✱ P✳ ❙❧❛t❡r✱ 1978 ✮ ✐t ❞♦❡s ❝♦♥t❛✐♥ ❡✈❡♥ ℓ ✕❝②❝❧❡s ✇❤❡r❡ ℓ = 6 , 8 , . . . , n ! ❤❛s ❤✐❡r❛r❝❤✐❝❛❧ str✉❝t✉r❡ ❤❛s ✐♥t❡❣r❛❧ s♣❡❝tr✉♠ ❊❧❡♥❛ ❑♦♥st❛♥t✐♥♦✈❛ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❙t❛r ❣r❛♣❤s ❙❏❚❯✲✷✵✶✽ ✷ ✴ ✸✺

❍✐❡r❛r❝❤✐❝❛❧ str✉❝t✉r❡ ♦❢ S n ❚❤❡ ❙t❛r ❣r❛♣❤ S n ✱ n � 3 ✱ ❝♦♥t❛✐♥s n ❝♦♣✐❡s S n − 1 ( i ) ✱ 1 � i � n ✱ ✇❤❡r❡ ❡❛❝❤ S n − 1 ✐s ❛♥ ✐♥❞✉❝❡❞ s✉❜❣r❛♣❤ ♦♥ t❤❡ ✈❡rt❡① s❡t V i = { [ π 1 . . . π n − 1 i ] , π k ∈ { 1 , . . . , n }\{ i } , 1 � k � n − 1 } ✱ | V i | = ( n − 1)! ❊①❛♠♣❧❡✿ S 4 ❤❛s ❢♦✉r ❝♦♣✐❡s ♦❢ S 3 ❊❧❡♥❛ ❑♦♥st❛♥t✐♥♦✈❛ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❙t❛r ❣r❛♣❤s ❙❏❚❯✲✷✵✶✽ ✸ ✴ ✸✺

■♥t❡❣r❛❧ ❣r❛♣❤ ❆ ❣r❛♣❤ ✐s ✐♥t❡❣r❛❧ ✐❢ ✐ts s♣❡❝tr✉♠ ❝♦♥s✐sts ❡♥t✐r❡❧② ♦❢ ✐♥t❡❣❡rs✳ ❋✳ ❍❛r❛r② ❛♥❞ ❆✳ ❏✳ ❙❝❤✇❡♥❦✱ ❲❤✐❝❤ ❣r❛♣❤s ❤❛✈❡ ✐♥t❡❣r❛❧ s♣❡❝tr❛❄ ●r❛♣❤s ❛♥❞ ❈♦♠❜✐♥✳ ✭✶✾✼✹✮ ■♥ ❋✳ ❍❛r❛r② ❛♥❞ ❆✳ ❏✳ ❙❝❤✇❡♥❦ ♣♦s❡❞ t❤❡ ♣r♦❜❧❡♠ ♦❢ ❝❤❛r❛❝t❡r✐③✐♥❣ t❤❡ ✐♥t❡❣r❛❧ ❣r❛♣❤s ✳ ❙♦♠❡ ❡①❛♠♣❧❡s ♦❢ ✐♥t❡❣r❛❧ ❣r❛♣❤s✿ ❝②❝❧❡s ✿ ✿ ✿ ❍✐st♦r✐❝❛❧ ❜❛❝❦❣r♦✉♥❞ ❙♣❡❝tr✉♠ ♦❢ ❛ ❣r❛♣❤ ❋♦r ❛ ❣r❛♣❤ Γ ✱ t❤❡ s♣❡❝tr✉♠ ♦❢ Γ ✐s t❤❡ s♣❡❝tr✉♠ ♦❢ ✐ts ❛❞❥❛❝❡♥❝② ♠❛tr✐①✳ ❊❧❡♥❛ ❑♦♥st❛♥t✐♥♦✈❛ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❙t❛r ❣r❛♣❤s ❙❏❚❯✲✷✵✶✽ ✹ ✴ ✸✺

❋✳ ❍❛r❛r② ❛♥❞ ❆✳ ❏✳ ❙❝❤✇❡♥❦✱ ❲❤✐❝❤ ❣r❛♣❤s ❤❛✈❡ ✐♥t❡❣r❛❧ s♣❡❝tr❛❄ ●r❛♣❤s ❛♥❞ ❈♦♠❜✐♥✳ ✭✶✾✼✹✮ ■♥ ❋✳ ❍❛r❛r② ❛♥❞ ❆✳ ❏✳ ❙❝❤✇❡♥❦ ♣♦s❡❞ t❤❡ ♣r♦❜❧❡♠ ♦❢ ❝❤❛r❛❝t❡r✐③✐♥❣ t❤❡ ✐♥t❡❣r❛❧ ❣r❛♣❤s ✳ ❙♦♠❡ ❡①❛♠♣❧❡s ♦❢ ✐♥t❡❣r❛❧ ❣r❛♣❤s✿ ❝②❝❧❡s ✿ ✿ ✿ ❍✐st♦r✐❝❛❧ ❜❛❝❦❣r♦✉♥❞ ❙♣❡❝tr✉♠ ♦❢ ❛ ❣r❛♣❤ ❋♦r ❛ ❣r❛♣❤ Γ ✱ t❤❡ s♣❡❝tr✉♠ ♦❢ Γ ✐s t❤❡ s♣❡❝tr✉♠ ♦❢ ✐ts ❛❞❥❛❝❡♥❝② ♠❛tr✐①✳ ■♥t❡❣r❛❧ ❣r❛♣❤ ❆ ❣r❛♣❤ Γ ✐s ✐♥t❡❣r❛❧ ✐❢ ✐ts s♣❡❝tr✉♠ ❝♦♥s✐sts ❡♥t✐r❡❧② ♦❢ ✐♥t❡❣❡rs✳ ❊❧❡♥❛ ❑♦♥st❛♥t✐♥♦✈❛ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❙t❛r ❣r❛♣❤s ❙❏❚❯✲✷✵✶✽ ✹ ✴ ✸✺

❙♦♠❡ ❡①❛♠♣❧❡s ♦❢ ✐♥t❡❣r❛❧ ❣r❛♣❤s✿ ❝②❝❧❡s ✿ ✿ ✿ ❍✐st♦r✐❝❛❧ ❜❛❝❦❣r♦✉♥❞ ❙♣❡❝tr✉♠ ♦❢ ❛ ❣r❛♣❤ ❋♦r ❛ ❣r❛♣❤ Γ ✱ t❤❡ s♣❡❝tr✉♠ ♦❢ Γ ✐s t❤❡ s♣❡❝tr✉♠ ♦❢ ✐ts ❛❞❥❛❝❡♥❝② ♠❛tr✐①✳ ■♥t❡❣r❛❧ ❣r❛♣❤ ❆ ❣r❛♣❤ Γ ✐s ✐♥t❡❣r❛❧ ✐❢ ✐ts s♣❡❝tr✉♠ ❝♦♥s✐sts ❡♥t✐r❡❧② ♦❢ ✐♥t❡❣❡rs✳ ❋✳ ❍❛r❛r② ❛♥❞ ❆✳ ❏✳ ❙❝❤✇❡♥❦✱ ❲❤✐❝❤ ❣r❛♣❤s ❤❛✈❡ ✐♥t❡❣r❛❧ s♣❡❝tr❛❄ ●r❛♣❤s ❛♥❞ ❈♦♠❜✐♥✳ ✭✶✾✼✹✮ ■♥ 1974 ❋✳ ❍❛r❛r② ❛♥❞ ❆✳ ❏✳ ❙❝❤✇❡♥❦ ♣♦s❡❞ t❤❡ ♣r♦❜❧❡♠ ♦❢ ❝❤❛r❛❝t❡r✐③✐♥❣ t❤❡ ✐♥t❡❣r❛❧ ❣r❛♣❤s ✳ ❊❧❡♥❛ ❑♦♥st❛♥t✐♥♦✈❛ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❙t❛r ❣r❛♣❤s ❙❏❚❯✲✷✵✶✽ ✹ ✴ ✸✺

❍✐st♦r✐❝❛❧ ❜❛❝❦❣r♦✉♥❞ ❙♣❡❝tr✉♠ ♦❢ ❛ ❣r❛♣❤ ❋♦r ❛ ❣r❛♣❤ Γ ✱ t❤❡ s♣❡❝tr✉♠ ♦❢ Γ ✐s t❤❡ s♣❡❝tr✉♠ ♦❢ ✐ts ❛❞❥❛❝❡♥❝② ♠❛tr✐①✳ ■♥t❡❣r❛❧ ❣r❛♣❤ ❆ ❣r❛♣❤ Γ ✐s ✐♥t❡❣r❛❧ ✐❢ ✐ts s♣❡❝tr✉♠ ❝♦♥s✐sts ❡♥t✐r❡❧② ♦❢ ✐♥t❡❣❡rs✳ ❋✳ ❍❛r❛r② ❛♥❞ ❆✳ ❏✳ ❙❝❤✇❡♥❦✱ ❲❤✐❝❤ ❣r❛♣❤s ❤❛✈❡ ✐♥t❡❣r❛❧ s♣❡❝tr❛❄ ●r❛♣❤s ❛♥❞ ❈♦♠❜✐♥✳ ✭✶✾✼✹✮ ■♥ 1974 ❋✳ ❍❛r❛r② ❛♥❞ ❆✳ ❏✳ ❙❝❤✇❡♥❦ ♣♦s❡❞ t❤❡ ♣r♦❜❧❡♠ ♦❢ ❝❤❛r❛❝t❡r✐③✐♥❣ t❤❡ ✐♥t❡❣r❛❧ ❣r❛♣❤s ✳ ❙♦♠❡ ❡①❛♠♣❧❡s ♦❢ ✐♥t❡❣r❛❧ ❣r❛♣❤s✿ ❝②❝❧❡s C 3 ✿ ( − 1 2 , 2) C 4 ✿ ( − 2 , 0 2 , 2) C 6 ✿ ( − 2 , − 1 2 , 1 2 , 2) ❊❧❡♥❛ ❑♦♥st❛♥t✐♥♦✈❛ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❙t❛r ❣r❛♣❤s ❙❏❚❯✲✷✵✶✽ ✹ ✴ ✸✺