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❆♥ ❡✐❣❡♥✈❛❧✉❡ ❜♦✉♥❞ ❢♦r t❤❡ ❑✐r❝❤❤♦✛✲▲❛♣❧❛❝✐❛♥ ♦♥ ♣❧❛♥❛r q✉❛♥t✉♠ ❣r❛♣❤s ▼❛r✈✐♥ P❧ü♠❡r ❯♥✐✈❡rs✐t② ♦❢ ❍❛❣❡♥ ❋❡❜r✉❛r② ✷✽✱ ✷✵✶✾ ▼❛r✈✐♥ P❧ü♠❡r ✭❯♥✐✈❡rs✐t② ♦❢ ❍❛❣❡♥✮ P❧❛♥❛r q✉❛♥t✉♠ ❣r❛♣❤s ❋❡❜r✉❛r② ✷✽✱ ✷✵✶✾ ✶ ✴ ✷✾

❲❡ s❤❛❧❧ ✇r✐t❡ ✐❢ ❛r❡ ❛❞❥❛❝❡♥t✳ ▲❡t ✐s ✐♥❝✐❞❡♥t t♦ ▲❡t ✵ ❞❡♥♦t❡ ❛ ♣♦s✐t✐✈❡ ✇❡✐❣❤t ❢✉♥❝t✐♦♥ ♦♥ t❤❡ ✈❡rt❡① s❡t✱ ❧❡t ❢♦r ✷ ❈♦♥s✐❞❡r t❤❡ s♣❛❝❡ ♦❢ ❢✉♥❝t✐♦♥s ❡q✉✐♣♣❡❞ ✇✐t❤ t❤❡ ♥♦r♠ ✷ ✷ ✷ ❚❤❡ ❝♦♠❜✐♥❛t♦r✐❛❧ s❡tt✐♥❣ ▲❡t G = ( V , E ) ❜❡ ❛ ✜♥✐t❡✱ s✐♠♣❧❡ ❛♥❞ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤ ✇✐t❤ ✈❡rt❡① s❡t V = V ( E ) ❛♥❞ ❡❞❣❡ s❡t E = E ( G ) ✳ ▼❛r✈✐♥ P❧ü♠❡r ✭❯♥✐✈❡rs✐t② ♦❢ ❍❛❣❡♥✮ P❧❛♥❛r q✉❛♥t✉♠ ❣r❛♣❤s ❋❡❜r✉❛r② ✷✽✱ ✷✵✶✾ ✷ ✴ ✷✾

▲❡t ✵ ❞❡♥♦t❡ ❛ ♣♦s✐t✐✈❡ ✇❡✐❣❤t ❢✉♥❝t✐♦♥ ♦♥ t❤❡ ✈❡rt❡① s❡t✱ ❧❡t ❢♦r ✷ ❈♦♥s✐❞❡r t❤❡ s♣❛❝❡ ♦❢ ❢✉♥❝t✐♦♥s ❡q✉✐♣♣❡❞ ✇✐t❤ t❤❡ ♥♦r♠ ✷ ✷ ✷ ❚❤❡ ❝♦♠❜✐♥❛t♦r✐❛❧ s❡tt✐♥❣ ▲❡t G = ( V , E ) ❜❡ ❛ ✜♥✐t❡✱ s✐♠♣❧❡ ❛♥❞ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤ ✇✐t❤ ✈❡rt❡① s❡t V = V ( E ) ❛♥❞ ❡❞❣❡ s❡t E = E ( G ) ✳ ❲❡ s❤❛❧❧ ✇r✐t❡ u ∼ v ✐❢ u , v ∈ V ❛r❡ ❛❞❥❛❝❡♥t✳ ▲❡t E v = { e ∈ E | e ✐s ✐♥❝✐❞❡♥t t♦ v } . ▼❛r✈✐♥ P❧ü♠❡r ✭❯♥✐✈❡rs✐t② ♦❢ ❍❛❣❡♥✮ P❧❛♥❛r q✉❛♥t✉♠ ❣r❛♣❤s ❋❡❜r✉❛r② ✷✽✱ ✷✵✶✾ ✷ ✴ ✷✾

✷ ❈♦♥s✐❞❡r t❤❡ s♣❛❝❡ ♦❢ ❢✉♥❝t✐♦♥s ❡q✉✐♣♣❡❞ ✇✐t❤ t❤❡ ♥♦r♠ ✷ ✷ ✷ ❚❤❡ ❝♦♠❜✐♥❛t♦r✐❛❧ s❡tt✐♥❣ ▲❡t G = ( V , E ) ❜❡ ❛ ✜♥✐t❡✱ s✐♠♣❧❡ ❛♥❞ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤ ✇✐t❤ ✈❡rt❡① s❡t V = V ( E ) ❛♥❞ ❡❞❣❡ s❡t E = E ( G ) ✳ ❲❡ s❤❛❧❧ ✇r✐t❡ u ∼ v ✐❢ u , v ∈ V ❛r❡ ❛❞❥❛❝❡♥t✳ ▲❡t E v = { e ∈ E | e ✐s ✐♥❝✐❞❡♥t t♦ v } . ▲❡t m : V → ( ✵ , ∞ ) ❞❡♥♦t❡ ❛ ♣♦s✐t✐✈❡ ✇❡✐❣❤t ❢✉♥❝t✐♦♥ ♦♥ t❤❡ ✈❡rt❡① s❡t✱ ❧❡t � m ( U ) := m ( u ) ❢♦r U ⊂ V . u ∈ U ▼❛r✈✐♥ P❧ü♠❡r ✭❯♥✐✈❡rs✐t② ♦❢ ❍❛❣❡♥✮ P❧❛♥❛r q✉❛♥t✉♠ ❣r❛♣❤s ❋❡❜r✉❛r② ✷✽✱ ✷✵✶✾ ✷ ✴ ✷✾

❚❤❡ ❝♦♠❜✐♥❛t♦r✐❛❧ s❡tt✐♥❣ ▲❡t G = ( V , E ) ❜❡ ❛ ✜♥✐t❡✱ s✐♠♣❧❡ ❛♥❞ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤ ✇✐t❤ ✈❡rt❡① s❡t V = V ( E ) ❛♥❞ ❡❞❣❡ s❡t E = E ( G ) ✳ ❲❡ s❤❛❧❧ ✇r✐t❡ u ∼ v ✐❢ u , v ∈ V ❛r❡ ❛❞❥❛❝❡♥t✳ ▲❡t E v = { e ∈ E | e ✐s ✐♥❝✐❞❡♥t t♦ v } . ▲❡t m : V → ( ✵ , ∞ ) ❞❡♥♦t❡ ❛ ♣♦s✐t✐✈❡ ✇❡✐❣❤t ❢✉♥❝t✐♦♥ ♦♥ t❤❡ ✈❡rt❡① s❡t✱ ❧❡t � m ( U ) := m ( u ) ❢♦r U ⊂ V . u ∈ U m ( V ; C d ) ♦❢ ❢✉♥❝t✐♦♥s f : V → C d ❡q✉✐♣♣❡❞ ✇✐t❤ t❤❡ ❈♦♥s✐❞❡r t❤❡ s♣❛❝❡ l ✷ ♥♦r♠ || f || ✷ � m ( u ) | f ( u ) | ✷ . l ✷ ( V ; C d ) = v ∈ V ▼❛r✈✐♥ P❧ü♠❡r ✭❯♥✐✈❡rs✐t② ♦❢ ❍❛❣❡♥✮ P❧❛♥❛r q✉❛♥t✉♠ ❣r❛♣❤s ❋❡❜r✉❛r② ✷✽✱ ✷✵✶✾ ✷ ✴ ✷✾

✷ ❖♥ ✇❡ ❝♦♥s✐❞❡r t❤❡ ✭✇❡✐❣❤t❡❞✮ ❝♦♠❜✐♥❛t♦r✐❛❧ ▲❛♣❧❛❝✐❛♥ ✱ ✶ ❛♥❞ ✐ts ❛ss♦❝✐❛t❡❞ q✉❛❞r❛t✐❝ ❢♦r♠ ❣✐✈❡♥ ❜② ✷ ❚❤❡ ❝♦♠❜✐♥❛t♦r✐❛❧ s❡tt✐♥❣ ▲❡t µ : E → ( ✵ , ∞ ) ❜❡ ❛ ♣♦s✐t✐✈❡ ❡❞❣❡ ✇❡✐❣❤t✳ ❚❤❡ ✭✇❡✐❣❤t❡❞✮ ❞❡❣r❡❡ ♦❢ ❛ ✈❡rt❡① v ∈ V ✇✐t❤ r❡s♣❡❝t t♦ µ ✐s � d µ v = µ ( e ) , d µ max = max v ∈ V d µ ✭✶✮ v e ∈ E v ▼❛r✈✐♥ P❧ü♠❡r ✭❯♥✐✈❡rs✐t② ♦❢ ❍❛❣❡♥✮ P❧❛♥❛r q✉❛♥t✉♠ ❣r❛♣❤s ❋❡❜r✉❛r② ✷✽✱ ✷✵✶✾ ✸ ✴ ✷✾

❛♥❞ ✐ts ❛ss♦❝✐❛t❡❞ q✉❛❞r❛t✐❝ ❢♦r♠ ❣✐✈❡♥ ❜② ✷ ❚❤❡ ❝♦♠❜✐♥❛t♦r✐❛❧ s❡tt✐♥❣ ▲❡t µ : E → ( ✵ , ∞ ) ❜❡ ❛ ♣♦s✐t✐✈❡ ❡❞❣❡ ✇❡✐❣❤t✳ ❚❤❡ ✭✇❡✐❣❤t❡❞✮ ❞❡❣r❡❡ ♦❢ ❛ ✈❡rt❡① v ∈ V ✇✐t❤ r❡s♣❡❝t t♦ µ ✐s � d µ v = µ ( e ) , d µ max = max v ∈ V d µ ✭✶✮ v e ∈ E v ❖♥ l ✷ m ( V ) ✇❡ ❝♦♥s✐❞❡r t❤❡ ✭✇❡✐❣❤t❡❞✮ ❝♦♠❜✐♥❛t♦r✐❛❧ ▲❛♣❧❛❝✐❛♥ L ✱ ✶ � ( L f )( u ) = µ ( e )( f ( u ) − f ( v )) , u ∈ V m ( u ) e = { u , v }∈ E v ▼❛r✈✐♥ P❧ü♠❡r ✭❯♥✐✈❡rs✐t② ♦❢ ❍❛❣❡♥✮ P❧❛♥❛r q✉❛♥t✉♠ ❣r❛♣❤s ❋❡❜r✉❛r② ✷✽✱ ✷✵✶✾ ✸ ✴ ✷✾

❚❤❡ ❝♦♠❜✐♥❛t♦r✐❛❧ s❡tt✐♥❣ ▲❡t µ : E → ( ✵ , ∞ ) ❜❡ ❛ ♣♦s✐t✐✈❡ ❡❞❣❡ ✇❡✐❣❤t✳ ❚❤❡ ✭✇❡✐❣❤t❡❞✮ ❞❡❣r❡❡ ♦❢ ❛ ✈❡rt❡① v ∈ V ✇✐t❤ r❡s♣❡❝t t♦ µ ✐s � d µ v = µ ( e ) , d µ max = max v ∈ V d µ ✭✶✮ v e ∈ E v ❖♥ l ✷ m ( V ) ✇❡ ❝♦♥s✐❞❡r t❤❡ ✭✇❡✐❣❤t❡❞✮ ❝♦♠❜✐♥❛t♦r✐❛❧ ▲❛♣❧❛❝✐❛♥ L ✱ ✶ � ( L f )( u ) = µ ( e )( f ( u ) − f ( v )) , u ∈ V m ( u ) e = { u , v }∈ E v ❛♥❞ ✐ts ❛ss♦❝✐❛t❡❞ q✉❛❞r❛t✐❝ ❢♦r♠ q ❣✐✈❡♥ ❜② � µ ( e ) | f ( u ) − f ( v ) | ✷ . q ( f ) = e = { u , v }∈ E ▼❛r✈✐♥ P❧ü♠❡r ✭❯♥✐✈❡rs✐t② ♦❢ ❍❛❣❡♥✮ P❧❛♥❛r q✉❛♥t✉♠ ❣r❛♣❤s ❋❡❜r✉❛r② ✷✽✱ ✷✵✶✾ ✸ ✴ ✷✾

❚❤❡ s♣❡❝tr❛❧ ❣❛♣ ♦❢ t❤❡ ❝♦♠❜✐♥❛t♦r✐❛❧ ▲❛♣❧❛❝✐❛♥ ▲❡♠♠❛ ❚❤❡ ✜rst ♣♦s✐t✐✈❡ ❡✐❣❡♥✈❛❧✉❡ ♦❢ L ✐s ❣✐✈❡♥ ❜② q ( f ) λ ✶ ( L ) = inf , || f || ✷ f ∈ l ✷ m ( V ) \{ ✵ } , l ✷ m ( V ) f ⊥ m ✶ V ✇❤❡r❡ � f ⊥ m ✶ V ⇔ m ( v ) f ( v ) = ✵ . v ∈ V ▼❛r✈✐♥ P❧ü♠❡r ✭❯♥✐✈❡rs✐t② ♦❢ ❍❛❣❡♥✮ P❧❛♥❛r q✉❛♥t✉♠ ❣r❛♣❤s ❋❡❜r✉❛r② ✷✽✱ ✷✵✶✾ ✹ ✴ ✷✾

❚❤❡ s♣❡❝tr❛❧ ❣❛♣ ♦❢ t❤❡ ❝♦♠❜✐♥❛t♦r✐❛❧ ▲❛♣❧❛❝✐❛♥ ▲❡♠♠❛ ❚❤❡ ✜rst ♣♦s✐t✐✈❡ ❡✐❣❡♥✈❛❧✉❡ ♦❢ L ✐s ❣✐✈❡♥ ❜② q ( f ) λ ✶ ( L ) = inf , || f || ✷ f ∈ l ✷ m ( V ; C d ) \{ ✵ } , l ✷ m ( V ; C d ) f ⊥ m ✶ V ✇❤❡r❡ � f ⊥ m ✶ V ⇔ m ( v ) f ( v ) = ✵ . v ∈ V ▼❛r✈✐♥ P❧ü♠❡r ✭❯♥✐✈❡rs✐t② ♦❢ ❍❛❣❡♥✮ P❧❛♥❛r q✉❛♥t✉♠ ❣r❛♣❤s ❋❡❜r✉❛r② ✷✽✱ ✷✵✶✾ ✺ ✴ ✷✾

❲❡ ✐♥tr♦❞✉❝❡ t❤❡ ❢♦❧❧♦✇✐♥❣ q✉❛♥t✐t✐❡s✿ t❤❡ t♦t❛❧ ❧❡♥❣t❤ ✱ t❤❡ ✭✇❡✐❣❤t❡❞✮ ❞❡❣r❡❡ ❢♦r ✳ ❚❤❡ ♠❡tr✐❝ s❡tt✐♥❣ ❈♦♥s✐❞❡r ❛ ♠❡tr✐❝ ❣r❛♣❤ G = ( G , l ) , ✇❤❡r❡ G = ( V , E ) ✐s ❛ ✭❝♦♠❜✐♥❛t♦r✐❛❧✮ ❝♦♥♥❡❝t❡❞✱ s✐♠♣❧❡ ❛♥❞ ✜♥✐t❡ ❣r❛♣❤ ❛♥❞ l : E → ( ✵ , ∞ ) , e �→ l e ✐s ❛ ✇❡✐❣❤t ❢✉♥❝t✐♦♥ ♦♥ t❤❡ ❡❞❣❡ s❡t✳ ▼❛r✈✐♥ P❧ü♠❡r ✭❯♥✐✈❡rs✐t② ♦❢ ❍❛❣❡♥✮ P❧❛♥❛r q✉❛♥t✉♠ ❣r❛♣❤s ❋❡❜r✉❛r② ✷✽✱ ✷✵✶✾ ✻ ✴ ✷✾