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●❡♥❡r❛❧✐③❡❞ ❈❛②❧❡② ❣r❛♣❤s ❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ✭❯♥✐✈❡rs✐t② ♦❢ Pr✐♠♦rs❦❛✮ ❏♦✐♥t ✇♦r❦ ✇✐t❤ ❑❧❛✈❞✐❥❛ ❑✉t♥❛r✱ P❛✇❡➟ P❡t❡❝❦✐ ❛♥❞ ❆♥❛st❛s✐②❛ ❚❛♥❛♥❛✳ ■♥t❡r♥❛t✐♦♥❛❧ ❈♦♥❢❡r❡♥❝❡ ♦♥ ●r❛♣❤ ❚❤❡♦r② ✷✼✳✺✳✷✵✶✺ ❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ●❡♥❡r❛❧✐③❡❞ ❈❛②❧❡② ❣r❛♣❤s

❖✈❡r✈✐❡✇ ❈❛②❧❡② ❣r❛♣❤s ●❡♥❡r❛❧✐③❡❞ ❈❛②❧❡② ❣r❛♣❤s ❆✉t♦♠♦r♣❤✐s♠ ♦❢ ●❡♥❡r❛❧✐③❡❞ ❈❛②❧❡② ❣r❛♣❤s ◆♦♥✲❈❛②❧❡② ✈❡rt❡①✲tr❛♥s✐t✐✈❡ ❣❡♥❡r❛❧✐③❡❞ ❈❛②❧❡② ❣r❛♣❤s ❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ●❡♥❡r❛❧✐③❡❞ ❈❛②❧❡② ❣r❛♣❤s

❇❛s✐❝s ❛❜♦✉t ❣r❛♣❤s ❆ ❣r❛♣❤ ✐s ❛♥ ♦r❞❡r❡❞ ♣❛✐r Γ = ( V , E ) ✱ ✇❤❡r❡ V ❞❡♥♦t❡s t❤❡ s❡t ♦❢ ✈❡rt✐❝❡s✱ ❛♥❞ E ❞❡♥♦t❡s t❤❡ s❡t ♦❢ ❡❞❣❡s ♦❢ t❤❡ ❣r❛♣❤ Γ . ❆✉t♦♠♦r♣❤✐s♠ ♦❢ ❛ ❣r❛♣❤ Γ ✐s ❛ ❜✐❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥ ϕ : V (Γ) → V (Γ) s✉❝❤ t❤❛t { x , y } ∈ E (Γ) ⇔ { ϕ ( x ) , ϕ ( y ) } ∈ E (Γ) . ❲❡ ❞❡✜♥❡ t❤❡ s❡t Aut (Γ) t♦ ❜❡ t❤❡ s❡t ♦❢ ❛❧❧ ❛✉t♦♠♦r♣❤✐s♠s ♦❢ t❤❡ ❣r❛♣❤ Γ . ■t ✐s ♥♦t ❞✐✣❝✉❧t t♦ s❡❡ t❤❛t Aut (Γ) ✐s ✐♥ ❢❛❝t t❤❡ ❣r♦✉♣ ✇✐t❤ r❡s♣❡❝t t♦ ❝♦♠♣♦s✐t✐♦♥ ♦❢ ❢✉♥❝t✐♦♥s✱ ❛♥❞ ✐t ✐s ❝❛❧❧❡❞ t❤❡ ❛✉t♦♠♦r♣❤✐s♠ ❣r♦✉♣ ♦❢ Γ . ❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ●❡♥❡r❛❧✐③❡❞ ❈❛②❧❡② ❣r❛♣❤s

❲❡ s❛② t❤❛t ❛ ❣r❛♣❤ ✐s ✈❡rt❡①✲tr❛♥s✐t✐✈❡ ❛❝ts tr❛♥s✐t✐✈❡❧② ♦♥ t❤❡ ✈❡rt❡① s❡t ❆✉t ♦❢ t❤❡ ❣r❛♣❤❀ ❡❞❣❡✲tr❛♥s✐t✐✈❡ ❛❝ts tr❛♥s✐t✐✈❡❧② ♦♥ t❤❡ ❡❞❣❡ s❡t ♦❢ ❆✉t t❤❡ ❣r❛♣❤❀ ❛r❝✲tr❛♥s✐t✐✈❡ ❆✉t ❛❝ts tr❛♥s✐t✐✈❡❧② ♦♥ t❤❡ ❛r❝ s❡t ♦❢ t❤❡ ❣r❛♣❤✳ ❆❝t✐♦♥s ♦♥ ❣r❛♣❤s ❲❤❡♥ ✇❡ s♣❡❛❦ ❛❜♦✉t ❣r❛♣❤s ✇❡ ❝♦♥s✐❞❡r t❤❡ ❛❝t✐♦♥ ♦❢ t❤❡ ❣r♦✉♣ ♦❢ t❤❡ ❛✉t♦♠♦r♣❤✐s♠s ♦♥ t❤❡ ❣r❛♣❤✳ ❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ●❡♥❡r❛❧✐③❡❞ ❈❛②❧❡② ❣r❛♣❤s

❆❝t✐♦♥s ♦♥ ❣r❛♣❤s ❲❤❡♥ ✇❡ s♣❡❛❦ ❛❜♦✉t ❣r❛♣❤s ✇❡ ❝♦♥s✐❞❡r t❤❡ ❛❝t✐♦♥ ♦❢ t❤❡ ❣r♦✉♣ ♦❢ t❤❡ ❛✉t♦♠♦r♣❤✐s♠s ♦♥ t❤❡ ❣r❛♣❤✳ ❲❡ s❛② t❤❛t ❛ ❣r❛♣❤ Γ ✐s ✈❡rt❡①✲tr❛♥s✐t✐✈❡ ⇔ ❆✉t (Γ) ❛❝ts tr❛♥s✐t✐✈❡❧② ♦♥ t❤❡ ✈❡rt❡① s❡t ♦❢ t❤❡ ❣r❛♣❤❀ ❡❞❣❡✲tr❛♥s✐t✐✈❡ ⇔ ❆✉t (Γ) ❛❝ts tr❛♥s✐t✐✈❡❧② ♦♥ t❤❡ ❡❞❣❡ s❡t ♦❢ t❤❡ ❣r❛♣❤❀ ❛r❝✲tr❛♥s✐t✐✈❡ ⇔ ❆✉t (Γ) ❛❝ts tr❛♥s✐t✐✈❡❧② ♦♥ t❤❡ ❛r❝ s❡t ♦❢ t❤❡ ❣r❛♣❤✳ ❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ●❡♥❡r❛❧✐③❡❞ ❈❛②❧❡② ❣r❛♣❤s

❊①❛♠♣❧❡ ✶✵ ✱ ✶ ✺ ✳ ❈❛②❧❡② ❣r❛♣❤s ●✐✈❡♥ ❛ ❣r♦✉♣ G ❛♥❞ ❛ s✉❜s❡t S ♦❢ G s✉❝❤ t❤❛t✿ ✭✐✮ ✶ �∈ S ✱ ✭✐✐✮ S − ✶ = S ❀ t❤❡ ❈❛②❧❡② ❣r❛♣❤ ❈❛② ( G , S ) ♦❢ G r❡❧❛t✐✈❡ t♦ S ❤❛s ✈❡rt❡① s❡t G ❛♥❞ ❡❞❣❡s ♦❢ t❤❡ ❢♦r♠ { g , gs } ✇❤❡r❡ g ∈ G ❛♥❞ s ∈ S ✳ ❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ●❡♥❡r❛❧✐③❡❞ ❈❛②❧❡② ❣r❛♣❤s

❈❛②❧❡② ❣r❛♣❤s ●✐✈❡♥ ❛ ❣r♦✉♣ G ❛♥❞ ❛ s✉❜s❡t S ♦❢ G s✉❝❤ t❤❛t✿ ✭✐✮ ✶ �∈ S ✱ ✭✐✐✮ S − ✶ = S ❀ t❤❡ ❈❛②❧❡② ❣r❛♣❤ ❈❛② ( G , S ) ♦❢ G r❡❧❛t✐✈❡ t♦ S ❤❛s ✈❡rt❡① s❡t G ❛♥❞ ❡❞❣❡s ♦❢ t❤❡ ❢♦r♠ { g , gs } ✇❤❡r❡ g ∈ G ❛♥❞ s ∈ S ✳ ❊①❛♠♣❧❡ G = Z ✶✵ ✱ S = {± ✶ , ✺ } ✳ ❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ●❡♥❡r❛❧✐③❡❞ ❈❛②❧❡② ❣r❛♣❤s

❍♦✇❡✈❡r✱ ♥♦t ❡✈❡r② ✈❡rt❡①✲tr❛♥s✐t✐✈❡ ❣r❛♣❤ ✐s ❈❛②❧❡② ❣r❛♣❤✳ ❚❤❡ s♠❛❧❧❡st ✈❡rt❡①✲tr❛♥s✐t✐✈❡ ❣r❛♣❤ ✇❤✐❝❤ ✐s ♥♦t ❈❛②❧❡② ✐s t❤❡ P❡t❡rs❡♥ ❣r❛♣❤✳ ❈❛②❧❡② ❣r❛♣❤s ■❢ X = Cay ( G , S ) t❤❡♥ t❤❡ ❛❝t✐♦♥ ♦❢ G ♦♥ ✐ts❡❧❢ ❜② t❤❡ ❧❡❢t ♠✉❧t✐♣❧✐❝❛t✐♦♥ ✐♥❞✉❝❡s ❛ s✉❜❣r♦✉♣ ♦❢ t❤❡ ❛✉t♦♠♦r♣❤✐s♠ ❣r♦✉♣ ✇❤✐❝❤ ❛❝ts tr❛♥s✐t✐✈❡❧② ♦♥ ✈❡rt✐❝❡s✱ ❤❡♥❝❡ ❡✈❡r② ❈❛②❧❡② ❣r❛♣❤ ✐s ✈❡rt❡①✲tr❛♥s✐t✐✈❡✳ ❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ●❡♥❡r❛❧✐③❡❞ ❈❛②❧❡② ❣r❛♣❤s

❈❛②❧❡② ❣r❛♣❤s ■❢ X = Cay ( G , S ) t❤❡♥ t❤❡ ❛❝t✐♦♥ ♦❢ G ♦♥ ✐ts❡❧❢ ❜② t❤❡ ❧❡❢t ♠✉❧t✐♣❧✐❝❛t✐♦♥ ✐♥❞✉❝❡s ❛ s✉❜❣r♦✉♣ ♦❢ t❤❡ ❛✉t♦♠♦r♣❤✐s♠ ❣r♦✉♣ ✇❤✐❝❤ ❛❝ts tr❛♥s✐t✐✈❡❧② ♦♥ ✈❡rt✐❝❡s✱ ❤❡♥❝❡ ❡✈❡r② ❈❛②❧❡② ❣r❛♣❤ ✐s ✈❡rt❡①✲tr❛♥s✐t✐✈❡✳ ❍♦✇❡✈❡r✱ ♥♦t ❡✈❡r② ✈❡rt❡①✲tr❛♥s✐t✐✈❡ ❣r❛♣❤ ✐s ❈❛②❧❡② ❣r❛♣❤✳ ❚❤❡ s♠❛❧❧❡st ✈❡rt❡①✲tr❛♥s✐t✐✈❡ ❣r❛♣❤ ✇❤✐❝❤ ✐s ♥♦t ❈❛②❧❡② ✐s t❤❡ P❡t❡rs❡♥ ❣r❛♣❤✳ ❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ●❡♥❡r❛❧✐③❡❞ ❈❛②❧❡② ❣r❛♣❤s

●❡♥❡r❛❧✐③❡❞ ❈❛②❧❡② ❣r❛♣❤s ▲❡t G ❜❡ ❛ ✜♥✐t❡ ❣r♦✉♣✱ S ❛ ♥♦♥✲❡♠♣t② s✉❜s❡t ♦❢ G ❛♥❞ α ❛♥ ❛✉t♦♠♦r♣❤✐s♠ ♦❢ G s✉❝❤ t❤❛t t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥s ❛r❡ s❛t✐s✜❡❞✿ ✭✐✮ α ✷ = ✶✱ ✭✐✐✮ α ( g − ✶ ) g �∈ S ✱ ✭ ∀ g ∈ G ✮ ✭✐✐✐✮ α ( S − ✶ ) = S ✳ ❚❤❡♥ t❤❡ ❣❡♥❡r❛❧✐③❡❞ ❈❛②❧❡② ❣r❛♣❤ X = GC ( G , S , α ) ♦♥ G ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ♦r❞❡r❡❞ ♣❛✐r ( S , α ) ✐s ❛ ❣r❛♣❤ ✇✐t❤ ✈❡rt❡① s❡t G ✱ ❛♥❞ ❡❞❣❡s ♦❢ ❢♦r♠ { g , α ( g ) s } ✱ ✇❤❡r❡ g ∈ G ❛♥❞ s ∈ S ✳ ❊①❛♠♣❧❡ G = Z ✶✵ ✱ α ( x ) = − x ✱ S = {± ✶ , ✺ } ✳ ❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ●❡♥❡r❛❧✐③❡❞ ❈❛②❧❡② ❣r❛♣❤s

01 21 11 00 10 20 22 02 12 ❊①❛♠♣❧❡ ❊①❛♠♣❧❡ ▲❡t G = Z ✸ × Z ✸ ✱ S = { ( ✶ , ✵ ) , ( ✶ , ✶ ) , ( ✵ , ✷ ) , ( ✷ , ✷ ) } ❛♥❞ α : ( i , j ) �→ ( j , i ) ✳ ❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ●❡♥❡r❛❧✐③❡❞ ❈❛②❧❡② ❣r❛♣❤s

❉♦✉❜❧❡ ❝♦✈❡r ♦❢ ❛ ❣r❛♣❤ ✐s t❤❡ ❞✐r❡❝t ♣r♦❞✉❝t ✷ ✳ ❚❤✐s ♠❡❛♥s t❤❛t ✷ ❛♥❞ ❛❧❧ t❤❡ ❡❞❣❡s ♦❢ ❛r❡ ✵ ✶ ❛♥❞ ✶ ✵ ✇❤❡r❡ ✐s ❛♥ ❡❞❣❡ ✐♥ ✳ ❍✐st♦r② ♦❢ ❣❡♥❡r❛❧✐③❡❞ ❈❛②❧❡② ❣r❛♣❤s ❚❤❡ ❝♦♥❝❡♣t ♦❢ ❣❡♥❡r❛❧✐③❡❞ ❈❛②❧❡② ❣r❛♣❤s ✇❛s ✐♥tr♦❞✉❝❡❞ ❜② ▼❛r✉➨✐↔✱ ❙❝❛♣❡❧❧❛t♦ ❛♥❞ ❩❛❣❛❣❧✐❛ ❙❛❧✈✐ ✐♥ ✶✾✾✷✳ ❚❤❡② st✉❞✐❡❞ ♣r♦♣❡rt✐❡s ♦❢ s✉❝❤ ❣r❛♣❤s r❡❧❛t✐✈❡ t♦ ❞♦✉❜❧❡ ❝♦✈❡rs ♦❢ ❣r❛♣❤s ✭s♦♠❡t✐♠❡s ❝❛❧❧❡❞ ❜✐♣❛rt✐t❡ ❞♦✉❜❧❡ ❝♦✈❡r ♦r ❝❛♥♦♥✐❝❛❧ ❞♦✉❜❧❡ ❝♦✈❡r✮✳ ❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ●❡♥❡r❛❧✐③❡❞ ❈❛②❧❡② ❣r❛♣❤s