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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 ✷✽✱ ✷✵✶✾ ✶ ✴ ✷✾

❚❤❡ ❝♦♠❜✐♥❛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❡ ✐❢ ❛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♠

✷

✷

✷

▼❛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 Ev = {e ∈ E | e ✐s ✐♥❝✐❞❡♥t t♦ v}. ▲❡t ✵ ❞❡♥♦t❡ ❛ ♣♦s✐t✐✈❡ ✇❡✐❣❤t ❢✉♥❝t✐♦♥ ♦♥ t❤❡ ✈❡rt❡① s❡t✱ ❧❡t ❢♦r ❈♦♥s✐❞❡r t❤❡ s♣❛❝❡

✷

♦❢ ❢✉♥❝t✐♦♥s ❡q✉✐♣♣❡❞ ✇✐t❤ t❤❡ ♥♦r♠

✷

✷

✷

▼❛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 Ev = {e ∈ E | e ✐s ✐♥❝✐❞❡♥t t♦ v}. ▲❡t m : V → (✵, ∞) ❞❡♥♦t❡ ❛ ♣♦s✐t✐✈❡ ✇❡✐❣❤t ❢✉♥❝t✐♦♥ ♦♥ t❤❡ ✈❡rt❡① s❡t✱ ❧❡t m(U) :=

• u∈U

m(u) ❢♦r U ⊂ V . ❈♦♥s✐❞❡r t❤❡ s♣❛❝❡

✷

♦❢ ❢✉♥❝t✐♦♥s ❡q✉✐♣♣❡❞ ✇✐t❤ t❤❡ ♥♦r♠

✷

✷

✷

▼❛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 Ev = {e ∈ E | e ✐s ✐♥❝✐❞❡♥t t♦ v}. ▲❡t m : V → (✵, ∞) ❞❡♥♦t❡ ❛ ♣♦s✐t✐✈❡ ✇❡✐❣❤t ❢✉♥❝t✐♦♥ ♦♥ t❤❡ ✈❡rt❡① s❡t✱ ❧❡t m(U) :=

• u∈U

m(u) ❢♦r U ⊂ V . ❈♦♥s✐❞❡r t❤❡ s♣❛❝❡ l✷

m(V ; Cd) ♦❢ ❢✉♥❝t✐♦♥s f : V → Cd ❡q✉✐♣♣❡❞ ✇✐t❤ t❤❡

♥♦r♠ ||f ||✷

l✷(V ;Cd) =

• v∈V

m(u)|f (u)|✷.

▼❛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 dµ

v =

• e∈Ev

µ(e), dµ

max = max v∈V dµ v

✭✶✮ ❖♥

✷

✇❡ ❝♦♥s✐❞❡r t❤❡ ✭✇❡✐❣❤t❡❞✮ ❝♦♠❜✐♥❛t♦r✐❛❧ ▲❛♣❧❛❝✐❛♥ ✱ ✶ ❛♥❞ ✐ts ❛ss♦❝✐❛t❡❞ q✉❛❞r❛t✐❝ ❢♦r♠ ❣✐✈❡♥ ❜②

✷

▼❛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 dµ

v =

• e∈Ev

µ(e), dµ

max = max v∈V dµ v

✭✶✮ ❖♥ l✷

m(V ) ✇❡ ❝♦♥s✐❞❡r t❤❡ ✭✇❡✐❣❤t❡❞✮ ❝♦♠❜✐♥❛t♦r✐❛❧ ▲❛♣❧❛❝✐❛♥ L✱

(Lf )(u) = ✶ m(u)

• e={u,v}∈Ev

µ(e)(f (u) − f (v)), u ∈ V ❛♥❞ ✐ts ❛ss♦❝✐❛t❡❞ q✉❛❞r❛t✐❝ ❢♦r♠ ❣✐✈❡♥ ❜②

✷

▼❛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 dµ

v =

• e∈Ev

µ(e), dµ

max = max v∈V dµ v

✭✶✮ ❖♥ l✷

m(V ) ✇❡ ❝♦♥s✐❞❡r t❤❡ ✭✇❡✐❣❤t❡❞✮ ❝♦♠❜✐♥❛t♦r✐❛❧ ▲❛♣❧❛❝✐❛♥ L✱

(Lf )(u) = ✶ m(u)

• e={u,v}∈Ev

µ(e)(f (u) − f (v)), u ∈ V ❛♥❞ ✐ts ❛ss♦❝✐❛t❡❞ q✉❛❞r❛t✐❝ ❢♦r♠ q ❣✐✈❡♥ ❜② q(f ) =

• e={u,v}∈E

µ(e)|f (u) − f (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 λ✶(L) = inf

f ∈l✷

m(V )\{✵},

f ⊥m✶V

q(f ) ||f ||✷

l✷

m(V )

, ✇❤❡r❡ f ⊥m ✶V ⇔

• v∈V

m(v)f (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 λ✶(L) = inf

f ∈l✷

m(V ;Cd)\{✵},

f ⊥m✶V

q(f ) ||f ||✷

l✷

m(V ;Cd)

, ✇❤❡r❡ f ⊥m ✶V ⇔

• v∈V

m(v)f (v) = ✵.

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

❚❤❡ ♠❡tr✐❝ s❡tt✐♥❣

❈♦♥s✐❞❡r ❛ ♠❡tr✐❝ ❣r❛♣❤ G = (G, l), ✇❤❡r❡ G = (V , E) ✐s ❛ ✭❝♦♠❜✐♥❛t♦r✐❛❧✮ ❝♦♥♥❡❝t❡❞✱ s✐♠♣❧❡ ❛♥❞ ✜♥✐t❡ ❣r❛♣❤ ❛♥❞ l : E → (✵, ∞), e → le ✐s ❛ ✇❡✐❣❤t ❢✉♥❝t✐♦♥ ♦♥ t❤❡ ❡❞❣❡ s❡t✳ ❲❡ ✐♥tr♦❞✉❝❡ t❤❡ ❢♦❧❧♦✇✐♥❣ q✉❛♥t✐t✐❡s✿ t❤❡ t♦t❛❧ ❧❡♥❣t❤ ✱ t❤❡ ✭✇❡✐❣❤t❡❞✮ ❞❡❣r❡❡ ❢♦r ✳

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

❚❤❡ ♠❡tr✐❝ s❡tt✐♥❣

❈♦♥s✐❞❡r ❛ ♠❡tr✐❝ ❣r❛♣❤ G = (G, l), ✇❤❡r❡ G = (V , E) ✐s ❛ ✭❝♦♠❜✐♥❛t♦r✐❛❧✮ ❝♦♥♥❡❝t❡❞✱ s✐♠♣❧❡ ❛♥❞ ✜♥✐t❡ ❣r❛♣❤ ❛♥❞ l : E → (✵, ∞), e → le ✐s ❛ ✇❡✐❣❤t ❢✉♥❝t✐♦♥ ♦♥ t❤❡ ❡❞❣❡ s❡t✳ ❲❡ ✐♥tr♦❞✉❝❡ t❤❡ ❢♦❧❧♦✇✐♥❣ q✉❛♥t✐t✐❡s✿ t❤❡ t♦t❛❧ ❧❡♥❣t❤ L =

e∈E le✱

t❤❡ ✭✇❡✐❣❤t❡❞✮ ❞❡❣r❡❡ dl

v = e∈Ev le ❢♦r v ∈ V ✳

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

❚❤❡ ♠❡tr✐❝ s❡tt✐♥❣

❖♥ t❤❡ ❤✐❧❜❡rt s♣❛❝❡ L✷(G; Cd) :=

• e∈E

L✷(✵, le; Cd), ||f ||✷

L✷(G;Cd) =

• e∈E

||fe||✷

L✷(✵,le;Cd).

✇❡ ❝♦♥s✐❞❡r t❤❡ ❑✐r❝❤❤♦✛✲▲❛♣❧❛❝✐❛♥ ❣✐✈❡♥ ❜② ❞✷ ❞ ✷ ❢♦r

✷ ✵

s✉❝❤ t❤❛t ✐s ❝♦♥t✐♥✉♦✉s ✐♥ ✵ ❢♦r ❛❧❧

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

❚❤❡ ♠❡tr✐❝ s❡tt✐♥❣

❖♥ t❤❡ ❤✐❧❜❡rt s♣❛❝❡ L✷(G; Cd) :=

• e∈E

L✷(✵, le; Cd), ||f ||✷

L✷(G;Cd) =

• e∈E

||fe||✷

L✷(✵,le;Cd).

✇❡ ❝♦♥s✐❞❡r t❤❡ ❑✐r❝❤❤♦✛✲▲❛♣❧❛❝✐❛♥ −∆ ❣✐✈❡♥ ❜② (−∆f )e = − ❞✷ ❞x✷

e

fe, ❢♦r f ∈

• e∈E

H✷(✵, le), s✉❝❤ t❤❛t f ✐s ❝♦♥t✐♥✉♦✉s ✐♥ v

• v∈V f ′

e(v) = ✵

• ❢♦r ❛❧❧ v ∈ V .

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

❈♦♠♣❛r✐♥❣ t❤❡ s♣❡❝tr❛❧ ❣❛♣s

▲❡♠♠❛

❲❡ ❝❤♦♦s❡ t❤❡ ✇❡✐❣❤ts m(v) = dl

v =

• e∈Ev

le, µ(e) = ✶ le t❤❡♥ t❤❡ ✜rst ♣♦s✐t✐✈❡ ♣♦s✐t✐✈❡ ❡✐❣❡♥✈❛❧✉❡s ♦❢ t❤❡ r❡s♣❡❝t✐✈❡ ▲❛♣❧❛❝✐❛♥s s❛t✐s❢② t❤❡ ❡st✐♠❛t❡ λ✶(−∆) ≤ ✻ λ✶(L).

Pr♦♦❢

❈♦♥s✐❞❡r ❡❞❣❡✇✐s❡ ❛✣♥❡ ❢✉♥❝t✐♦♥s ❛♥❞ ❝♦♠♣❛r❡ t❤❡ ❘❛②❧❡✐❣❤ q✉♦t✐❡♥ts✳

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

❈♦♠♣❛r✐♥❣ t❤❡ s♣❡❝tr❛❧ ❣❛♣s

▲❡♠♠❛

❲❡ ❝❤♦♦s❡ t❤❡ ✇❡✐❣❤ts m(v) = dl

v =

• e∈Ev

le, µ(e) = ✶ le t❤❡♥ t❤❡ ✜rst ♣♦s✐t✐✈❡ ♣♦s✐t✐✈❡ ❡✐❣❡♥✈❛❧✉❡s ♦❢ t❤❡ r❡s♣❡❝t✐✈❡ ▲❛♣❧❛❝✐❛♥s s❛t✐s❢② t❤❡ ❡st✐♠❛t❡ λ✶(−∆) ≤ ✻ λ✶(L).

Pr♦♦❢

❈♦♥s✐❞❡r ❡❞❣❡✇✐s❡ ❛✣♥❡ ❢✉♥❝t✐♦♥s ❛♥❞ ❝♦♠♣❛r❡ t❤❡ ❘❛②❧❡✐❣❤ q✉♦t✐❡♥ts✳

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

❲❤❛t ✐s ❦♥♦✇♥ ✐♥ t❤❡ ♠❛♥✐❢♦❧❞ ❝❛s❡❄

❚❤❡♦r❡♠ ✭❍❛ss❛♥♥❡③❤❛❞ ✬✶✶✮

• ✐✈❡♥ ❛ ❝❧♦s❡❞✱ ♦r✐❡♥t❡❞ ❘✐❡♠❛♥♥✐❛♥ ♠❛♥✐❢♦❧❞ M ♦❢ ❣❡♥✉s g > ✵ ✇❡ ❤❛✈❡

λk(−∆M) ≤ C g + k Vol(M). Pr❡✈✐♦✉s r❡s✉❧ts✿ ❍❡rs❝❤ ✬✼✵✱ ❨❛♥❣✱ ❨❛✉ ✬✽✵✱ ❑♦r❡✈❛❛r ✬✾✸

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

❲❤❛t ✐s ❦♥♦✇♥ ✐♥ t❤❡ ✭❝❧❛ss✐❝❛❧✮ ❝♦♠❜✐♥❛t♦r✐❛❧ ❝❛s❡❄

(Lf )(u) =

• v∼u

(f (u) − f (v)), u ∈ V . (m ≡ ✶, µ ≡ ✶)

❚❤❡♦r❡♠ ✭❙♣✐❡❧♠❛♥✱ ❚❡♥❣ ✬✵✼✮

✶

✐❢ ✐s ♣❧❛♥❛r

❚❤❡♦r❡♠ ✭❑❡❧♥❡r ✬✵✻✮

✶

✐❢ ✐s ♦❢ ❣❡♥✉s ✵

❚❤❡♦r❡♠ ✭❆♠✐♥✐✱ ❈♦❤❡♥✲❙t❡✐♥❡r ✬✶✽✮

✷

✐❢ ✐s ♦❢ ❣❡♥✉s ✵

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

❲❤❛t ✐s ❦♥♦✇♥ ✐♥ t❤❡ ✭❝❧❛ss✐❝❛❧✮ ❝♦♠❜✐♥❛t♦r✐❛❧ ❝❛s❡❄

(Lf )(u) =

• v∼u

(f (u) − f (v)), u ∈ V . (m ≡ ✶, µ ≡ ✶)

❚❤❡♦r❡♠ ✭❙♣✐❡❧♠❛♥✱ ❚❡♥❣ ✬✵✼✮

λ✶(L) ≤ C dmax |V | , ✐❢ G ✐s ♣❧❛♥❛r.

❚❤❡♦r❡♠ ✭❑❡❧♥❡r ✬✵✻✮

✶

✐❢ ✐s ♦❢ ❣❡♥✉s ✵

❚❤❡♦r❡♠ ✭❆♠✐♥✐✱ ❈♦❤❡♥✲❙t❡✐♥❡r ✬✶✽✮

✷

✐❢ ✐s ♦❢ ❣❡♥✉s ✵

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

❲❤❛t ✐s ❦♥♦✇♥ ✐♥ t❤❡ ✭❝❧❛ss✐❝❛❧✮ ❝♦♠❜✐♥❛t♦r✐❛❧ ❝❛s❡❄

(Lf )(u) =

• v∼u

(f (u) − f (v)), u ∈ V . (m ≡ ✶, µ ≡ ✶)

❚❤❡♦r❡♠ ✭❙♣✐❡❧♠❛♥✱ ❚❡♥❣ ✬✵✼✮

λ✶(L) ≤ C dmax |V | , ✐❢ G ✐s ♣❧❛♥❛r.

❚❤❡♦r❡♠ ✭❑❡❧♥❡r ✬✵✻✮

λ✶(L) ≤ poly(dmax) g |V |, ✐❢ G ✐s ♦❢ ❣❡♥✉s g > ✵

❚❤❡♦r❡♠ ✭❆♠✐♥✐✱ ❈♦❤❡♥✲❙t❡✐♥❡r ✬✶✽✮

✷

✐❢ ✐s ♦❢ ❣❡♥✉s ✵

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

❲❤❛t ✐s ❦♥♦✇♥ ✐♥ t❤❡ ✭❝❧❛ss✐❝❛❧✮ ❝♦♠❜✐♥❛t♦r✐❛❧ ❝❛s❡❄

(Lf )(u) =

• v∼u

(f (u) − f (v)), u ∈ V . (m ≡ ✶, µ ≡ ✶)

❚❤❡♦r❡♠ ✭❙♣✐❡❧♠❛♥✱ ❚❡♥❣ ✬✵✼✮

λ✶(L) ≤ C dmax |V | , ✐❢ G ✐s ♣❧❛♥❛r.

❚❤❡♦r❡♠ ✭❑❡❧♥❡r ✬✵✻✮

λ✶(L) ≤ poly(dmax) g |V |, ✐❢ G ✐s ♦❢ ❣❡♥✉s g > ✵

❚❤❡♦r❡♠ ✭❆♠✐♥✐✱ ❈♦❤❡♥✲❙t❡✐♥❡r ✬✶✽✮

λk(L) ≤ C d✷

max(g + k)

|V | ✐❢ G ✐s ♦❢ ❣❡♥✉s g ≥ ✵

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

❈✐r❝❧❡✲P❛❝❦✐♥❣s ❢♦r P❧❛♥❛r ●r❛♣❤s ✐♥ t❤❡ ♣❧❛♥❡

❚❤❡ ♣r♦♦❢ ♦❢ ❙♣✐❡❧♠❛♥ ❛♥❞ ❚❡♥❣ ✉s❡❞ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ♣❧❛♥❛r ❣r❛♣❤s✳

❚❤❡♦r❡♠ ✭❑♦❡❜❡ ✬✸✻✱ ❆♥❞r❡❡✈ ✬✼✵✱ ❚❤✉rst♦♥ ✬✼✽✮

❆ ❣r❛♣❤ G = (V , E) ✐s ♣❧❛♥❛r✱ ✐✛ t❤❡r❡ ❡①✐sts ❛ ❢❛♠✐❧② ♦❢ ❝❧♦s❡❞ ❞✐s❦s (Dv)v∈V ✐♥ t❤❡ ♣❧❛♥❡✱ s✉❝❤ t❤❛t t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞s ❢♦r ❛♥② t✇♦ ✈❡rt✐❝❡s v = u✿ ■❢ v ❛♥❞ u ❛r❡ ❛❞❥❛❝❡♥t✱ t❤❡ t✇♦ ❞✐s❦s Dv ✉♥❞ Du ✐♥t❡rs❡❝t ❛t ❡①❛❝t❧② ♦♥❡ ♣♦✐♥t✳ ■❢ v ❛♥❞ u ❛r❡ ♥♦t ❛❞❥❛❝❡♥t✱ t❤❡ t✇♦ ❞✐s❦s Dv ❛♥❞ Du ❛r❡ ❞✐s❥♦✐♥t✳ ■♥ t❤❡ ❛❜♦✈❡ ❝❛s❡ (Dv)v∈V ✐s ❝❛❧❧❡❞ ❛ ❝✐r❝❧❡✲♣❛❝❦✐♥❣ ❢♦r G ✐♥ R✷✳

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

❊①❛♠♣❧❡ ♦❢ ❛ ❝✐r❝❧❡✲♣❛❝❦✐♥❣

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

❈❛♣s ♦♥ t❤❡ ✉♥✐t s♣❤❡r❡ S✷

❲❡ s❤❛❧❧ tr❛♥s❢❡r t❤❡ ❝♦♥❝❡♣t ♦❢ ❝✐r❝❧❡ ♣❛❝❦✐♥❣s t♦ t❤❡ ✉♥✐t s♣❤❡r❡✳ ❆ s✉❜s❡t k ⊂ S✷ ✐s ❝❛❧❧❡❞ ❛ ❝✐r❝✉❧❛r ❧✐♥❡✱ ✐❢ k ✐s ❛ ♥♦♥✲tr✐✈✐❛❧ ✐♥t❡rs❡❝t✐♦♥ ♦❢ t❤❡ s♣❤❡r❡ S✷ ❛♥❞ ❛ ❤②♣❡r♣❧❛♥❡ H ✐♥ R✸✳ ❆ ❝♦♥♥❡❝t❡❞✱ ❝❧♦s❡❞ s✉❜s❡t

✷ ✐s ❝❛❧❧❡❞ ✭s♣❤❡r✐❝❛❧✮ ❝❛♣✱ ✐❢ ✐ts

❜♦✉♥❞❛r② ✐s ❛ ❝✐r❝✉❧❛r ❧✐♥❡✳ ❚❤❡r❡ ✐s ❡①❛❝t❧② ♦♥❡ ♣♦✐♥t ♦❢ ❡q✉❛❧ ❡✉❝❧✐❞✐❛♥ ❞✐st❛♥❝❡ t♦ ❛♥② ♣♦✐♥t ♦♥ t❤❡ ❜♦✉♥❞❛r② ♦❢ ✳ ❲❡ ❝❛❧❧ t❤❡ ❝❡♥t❡r ❛♥❞ t❤❡ r❛❞✐✉s ♦❢ ✳ ❚❤❡ s✉r❢❛❝❡ ❛r❡❛ ♦❢ ✐s

✷✳

❋❛❝t

❚❤❡ st❡r❡♦❣r❛♣❤✐❝ ♣r♦❥❡❝t✐♦♥

✷ ✷ ♠❛♣s ❞✐s❦s ✐♥ ✷ t♦ ❝❛♣s ✐♥ ✷ ❛♥❞

✈✐❝❡ ✈❡rs❛✳

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

❈❛♣s ♦♥ t❤❡ ✉♥✐t s♣❤❡r❡ S✷

❲❡ s❤❛❧❧ tr❛♥s❢❡r t❤❡ ❝♦♥❝❡♣t ♦❢ ❝✐r❝❧❡ ♣❛❝❦✐♥❣s t♦ t❤❡ ✉♥✐t s♣❤❡r❡✳ ❆ s✉❜s❡t k ⊂ S✷ ✐s ❝❛❧❧❡❞ ❛ ❝✐r❝✉❧❛r ❧✐♥❡✱ ✐❢ k ✐s ❛ ♥♦♥✲tr✐✈✐❛❧ ✐♥t❡rs❡❝t✐♦♥ ♦❢ t❤❡ s♣❤❡r❡ S✷ ❛♥❞ ❛ ❤②♣❡r♣❧❛♥❡ H ✐♥ R✸✳ ❆ ❝♦♥♥❡❝t❡❞✱ ❝❧♦s❡❞ s✉❜s❡t C ⊂ S✷ ✐s ❝❛❧❧❡❞ ✭s♣❤❡r✐❝❛❧✮ ❝❛♣✱ ✐❢ ✐ts ❜♦✉♥❞❛r② ✐s ❛ ❝✐r❝✉❧❛r ❧✐♥❡✳ ❚❤❡r❡ ✐s ❡①❛❝t❧② ♦♥❡ ♣♦✐♥t ♦❢ ❡q✉❛❧ ❡✉❝❧✐❞✐❛♥ ❞✐st❛♥❝❡ t♦ ❛♥② ♣♦✐♥t ♦♥ t❤❡ ❜♦✉♥❞❛r② ♦❢ ✳ ❲❡ ❝❛❧❧ t❤❡ ❝❡♥t❡r ❛♥❞ t❤❡ r❛❞✐✉s ♦❢ ✳ ❚❤❡ s✉r❢❛❝❡ ❛r❡❛ ♦❢ ✐s

✷✳

❋❛❝t

❚❤❡ st❡r❡♦❣r❛♣❤✐❝ ♣r♦❥❡❝t✐♦♥

✷ ✷ ♠❛♣s ❞✐s❦s ✐♥ ✷ t♦ ❝❛♣s ✐♥ ✷ ❛♥❞

✈✐❝❡ ✈❡rs❛✳

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

❈❛♣s ♦♥ t❤❡ ✉♥✐t s♣❤❡r❡ S✷

❲❡ s❤❛❧❧ tr❛♥s❢❡r t❤❡ ❝♦♥❝❡♣t ♦❢ ❝✐r❝❧❡ ♣❛❝❦✐♥❣s t♦ t❤❡ ✉♥✐t s♣❤❡r❡✳ ❆ s✉❜s❡t k ⊂ S✷ ✐s ❝❛❧❧❡❞ ❛ ❝✐r❝✉❧❛r ❧✐♥❡✱ ✐❢ k ✐s ❛ ♥♦♥✲tr✐✈✐❛❧ ✐♥t❡rs❡❝t✐♦♥ ♦❢ t❤❡ s♣❤❡r❡ S✷ ❛♥❞ ❛ ❤②♣❡r♣❧❛♥❡ H ✐♥ R✸✳ ❆ ❝♦♥♥❡❝t❡❞✱ ❝❧♦s❡❞ s✉❜s❡t C ⊂ S✷ ✐s ❝❛❧❧❡❞ ✭s♣❤❡r✐❝❛❧✮ ❝❛♣✱ ✐❢ ✐ts ❜♦✉♥❞❛r② ✐s ❛ ❝✐r❝✉❧❛r ❧✐♥❡✳ ❚❤❡r❡ ✐s ❡①❛❝t❧② ♦♥❡ ♣♦✐♥t p(C) ∈ C ♦❢ ❡q✉❛❧ ❡✉❝❧✐❞✐❛♥ ❞✐st❛♥❝❡ r(C) t♦ ❛♥② ♣♦✐♥t ♦♥ t❤❡ ❜♦✉♥❞❛r② ♦❢ C✳ ❲❡ ❝❛❧❧ p(C) t❤❡ ❝❡♥t❡r ❛♥❞ r(C) t❤❡ r❛❞✐✉s ♦❢ C✳ ❚❤❡ s✉r❢❛❝❡ ❛r❡❛ ♦❢ ✐s

✷✳

❋❛❝t

❚❤❡ st❡r❡♦❣r❛♣❤✐❝ ♣r♦❥❡❝t✐♦♥

✷ ✷ ♠❛♣s ❞✐s❦s ✐♥ ✷ t♦ ❝❛♣s ✐♥ ✷ ❛♥❞

✈✐❝❡ ✈❡rs❛✳

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

❈❛♣s ♦♥ t❤❡ ✉♥✐t s♣❤❡r❡ S✷

❲❡ s❤❛❧❧ tr❛♥s❢❡r t❤❡ ❝♦♥❝❡♣t ♦❢ ❝✐r❝❧❡ ♣❛❝❦✐♥❣s t♦ t❤❡ ✉♥✐t s♣❤❡r❡✳ ❆ s✉❜s❡t k ⊂ S✷ ✐s ❝❛❧❧❡❞ ❛ ❝✐r❝✉❧❛r ❧✐♥❡✱ ✐❢ k ✐s ❛ ♥♦♥✲tr✐✈✐❛❧ ✐♥t❡rs❡❝t✐♦♥ ♦❢ t❤❡ s♣❤❡r❡ S✷ ❛♥❞ ❛ ❤②♣❡r♣❧❛♥❡ H ✐♥ R✸✳ ❆ ❝♦♥♥❡❝t❡❞✱ ❝❧♦s❡❞ s✉❜s❡t C ⊂ S✷ ✐s ❝❛❧❧❡❞ ✭s♣❤❡r✐❝❛❧✮ ❝❛♣✱ ✐❢ ✐ts ❜♦✉♥❞❛r② ✐s ❛ ❝✐r❝✉❧❛r ❧✐♥❡✳ ❚❤❡r❡ ✐s ❡①❛❝t❧② ♦♥❡ ♣♦✐♥t p(C) ∈ C ♦❢ ❡q✉❛❧ ❡✉❝❧✐❞✐❛♥ ❞✐st❛♥❝❡ r(C) t♦ ❛♥② ♣♦✐♥t ♦♥ t❤❡ ❜♦✉♥❞❛r② ♦❢ C✳ ❲❡ ❝❛❧❧ p(C) t❤❡ ❝❡♥t❡r ❛♥❞ r(C) t❤❡ r❛❞✐✉s ♦❢ C✳ ❚❤❡ s✉r❢❛❝❡ ❛r❡❛ ♦❢ C ✐s π · r(C)✷✳

❋❛❝t

❚❤❡ st❡r❡♦❣r❛♣❤✐❝ ♣r♦❥❡❝t✐♦♥

✷ ✷ ♠❛♣s ❞✐s❦s ✐♥ ✷ t♦ ❝❛♣s ✐♥ ✷ ❛♥❞

✈✐❝❡ ✈❡rs❛✳

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

❈❛♣s ♦♥ t❤❡ ✉♥✐t s♣❤❡r❡ S✷

❲❡ s❤❛❧❧ tr❛♥s❢❡r t❤❡ ❝♦♥❝❡♣t ♦❢ ❝✐r❝❧❡ ♣❛❝❦✐♥❣s t♦ t❤❡ ✉♥✐t s♣❤❡r❡✳ ❆ s✉❜s❡t k ⊂ S✷ ✐s ❝❛❧❧❡❞ ❛ ❝✐r❝✉❧❛r ❧✐♥❡✱ ✐❢ k ✐s ❛ ♥♦♥✲tr✐✈✐❛❧ ✐♥t❡rs❡❝t✐♦♥ ♦❢ t❤❡ s♣❤❡r❡ S✷ ❛♥❞ ❛ ❤②♣❡r♣❧❛♥❡ H ✐♥ R✸✳ ❆ ❝♦♥♥❡❝t❡❞✱ ❝❧♦s❡❞ s✉❜s❡t C ⊂ S✷ ✐s ❝❛❧❧❡❞ ✭s♣❤❡r✐❝❛❧✮ ❝❛♣✱ ✐❢ ✐ts ❜♦✉♥❞❛r② ✐s ❛ ❝✐r❝✉❧❛r ❧✐♥❡✳ ❚❤❡r❡ ✐s ❡①❛❝t❧② ♦♥❡ ♣♦✐♥t p(C) ∈ C ♦❢ ❡q✉❛❧ ❡✉❝❧✐❞✐❛♥ ❞✐st❛♥❝❡ r(C) t♦ ❛♥② ♣♦✐♥t ♦♥ t❤❡ ❜♦✉♥❞❛r② ♦❢ C✳ ❲❡ ❝❛❧❧ p(C) t❤❡ ❝❡♥t❡r ❛♥❞ r(C) t❤❡ r❛❞✐✉s ♦❢ C✳ ❚❤❡ s✉r❢❛❝❡ ❛r❡❛ ♦❢ C ✐s π · r(C)✷✳

❋❛❝t

❚❤❡ st❡r❡♦❣r❛♣❤✐❝ ♣r♦❥❡❝t✐♦♥ R✷ → S✷ ♠❛♣s ❞✐s❦s ✐♥ R✷ t♦ ❝❛♣s ✐♥ S✷ ❛♥❞ ✈✐❝❡ ✈❡rs❛✳

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

❈✐r❝❧❡✲P❛❝❦✐♥❣s ❢♦r P❧❛♥❛r ●r❛♣❤s ✐♥ t❤❡ ♣❧❛♥❡

❚❤❡♦r❡♠ ✭❑♦❡❜❡ ✬✸✻✱ ❆♥❞r❡❡✈ ✬✼✵✱ ❚❤✉rst♦♥ ✬✼✽✮

❆ ❣r❛♣❤ G = (V , E) ✐s ♣❧❛♥❛r✱ ✐✛ t❤❡r❡ ❡①✐sts ❛ ❢❛♠✐❧② ♦❢ ❝❧♦s❡❞ ❞✐s❦s (Dv)v∈V ✱ s✉❝❤ t❤❛t t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞s ❢♦r ❛♥② t✇♦ ✈❡rt✐❝❡s v = u✿

✶ ■❢ v ❛♥❞ u ❛r❡ ❛❞❥❛❝❡♥t✱ t❤❡ t✇♦ ❞✐s❦s Dv ✉♥❞ Du ✐♥t❡rs❡❝t ❛t ❡①❛❝t❧②

♦♥❡ ♣♦✐♥t✳

✷ ■❢ v ❛♥❞ u ❛r❡ ♥♦t ❛❞❥❛❝❡♥t✱ t❤❡ t✇♦ ❞✐s❦s Dv ❛♥❞ Du ❛r❡ ❞✐s❥♦✐♥t✳

■♥ t❤❡ ❛❜♦✈❡ ❝❛s❡ (Dv)v∈V ✐s ❝❛❧❧❡❞ ❛ ❝✐r❝❧❡✲♣❛❝❦✐♥❣ ❢♦r G ✐♥ R✷✳

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

❈✐r❝❧❡✲P❛❝❦✐♥❣s ❢♦r P❧❛♥❛r ●r❛♣❤s ✐♥ t❤❡ ♣❧❛♥❡

❚❤❡♦r❡♠ ✭❑♦❡❜❡ ✬✸✻✱ ❆♥❞r❡❡✈ ✬✼✵✱ ❚❤✉rst♦♥ ✬✼✽✮

❆ ❣r❛♣❤ G = (V , E) ✐s ♣❧❛♥❛r✱ ✐✛ t❤❡r❡ ❡①✐sts ❛ ❢❛♠✐❧② ♦❢ ❝❧♦s❡❞ ❞✐s❦s (Dv)v∈V ✱ s✉❝❤ t❤❛t t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞s ❢♦r ❛♥② t✇♦ ✈❡rt✐❝❡s v = u✿

✶ ■❢ v ❛♥❞ u ❛r❡ ❛❞❥❛❝❡♥t✱ t❤❡ t✇♦ ❞✐s❦s Dv ✉♥❞ Du ✐♥t❡rs❡❝t ❛t ❡①❛❝t❧②

♦♥❡ ♣♦✐♥t✳

✷ ■❢ v ❛♥❞ u ❛r❡ ♥♦t ❛❞❥❛❝❡♥t✱ t❤❡ t✇♦ ❞✐s❦s Dv ✉♥❞ Du ❛r❡ ❞✐s❥♦✐♥t✳

■♥ t❤❡ ❛❜♦✈❡ ❝❛s❡ (Dv)v∈V ✐s ❝❛❧❧❡❞ ❛ ❝✐r❝❧❡✲♣❛❝❦✐♥❣ ❢♦r G ✐♥ R✷✳

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

❈✐r❝❧❡✲P❛❝❦✐♥❣s ❢♦r P❧❛♥❛r ●r❛♣❤s ✐♥ t❤❡ s♣❤❡r❡

❚❤❡♦r❡♠ ✭❑♦❡❜❡ ✬✸✻✱ ❆♥❞r❡❡✈ ✬✼✵✱ ❚❤✉rst♦♥ ✬✼✽✮

❆ ❣r❛♣❤ G = (V , E) ✐s ♣❧❛♥❛r✱ ✐✛ t❤❡r❡ ❡①✐sts ❛ ❢❛♠✐❧② ♦❢ s♣❤❡r✐❝❛❧ ❝❛♣s (Cv)v∈V ✱ s✉❝❤ t❤❛t t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞s ❢♦r ❛♥② t✇♦ ✈❡rt✐❝❡s v = u✿

✶ ■❢ v ❛♥❞ u ❛r❡ ❛❞❥❛❝❡♥t✱ t❤❡ t✇♦ ❝❛♣s Cv ✉♥❞ Cu ✐♥t❡rs❡❝t ❛t ❡①❛❝t❧②

♦♥❡ ♣♦✐♥t✳

✷ ■❢ v ❛♥❞ u ❛r❡ ♥♦t ❛❞❥❛❝❡♥t✱ t❤❡ t✇♦ ❝❛♣s Cv ❛♥❞ Cu ❛r❡ ❞✐s❥♦✐♥t✳

■♥ t❤❡ ❛❜♦✈❡ ❝❛s❡ (Cv)v∈V ✐s ❝❛❧❧❡❞ ❛ ❝✐r❝❧❡✲♣❛❝❦✐♥❣ ❢♦r G ✐♥ S✷✳

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

❆ ✜rst ❡✐❣❡♥✈❛❧✉❡ ❜♦✉♥❞ ❢♦r t❤❡ ❝♦♠❜✐♥❛t♦r✐❛❧ ▲❛♣❧❛❝✐❛♥

▲❡♠♠❛

❆ss✉♠❡ G ✐s ♣❧❛♥❛r ❛♥❞ t❤❡r❡ ✐s ❛ ❝✐r❝❧❡✲♣❛❝❦✐♥❣ (Cv)v∈V ❢♦r G✱ s✉❝❤ t❤❛t

• v∈V

m(v)p(Cv) = ✵, t❤❡♥ λ✶(L) ≤ ✽ dµ

max

m(V ) ❤♦❧❞s✳

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

Pr♦♦❢ ♦❢ t❤❡ ▲❡♠♠❛

■❞❡❛

❯s❡ ❛ ❝✐r❝❧❡✲♣❛❝❦✐♥❣ (Cv)v∈V r❡♣r❡s❡♥t✐♥❣ t❤❡ ♣❧❛♥❛r ❣r❛♣❤ G t♦ ♦❜t❛✐♥ ❛ t❡st ❢✉♥❝t✐♦♥ f ✐♥ t❤❡ ✈❡❝t♦r✲✈❛❧✉❡❞ ✈❛r✐❛t✐♦♥❛❧ ❢♦r♠✉❧❛t✐♦♥✳ ▲❡t ❛♥❞ ❢♦r ✳ ❲❡ ❝♦♥s✐❞❡r

✷ ✸ ❣✐✈❡♥ ❜②

❢♦r ✳ ❚❤❡♥

✷

✷ ✸

✷

.

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

Pr♦♦❢ ♦❢ t❤❡ ▲❡♠♠❛

■❞❡❛

❯s❡ ❛ ❝✐r❝❧❡✲♣❛❝❦✐♥❣ (Cv)v∈V r❡♣r❡s❡♥t✐♥❣ t❤❡ ♣❧❛♥❛r ❣r❛♣❤ G t♦ ♦❜t❛✐♥ ❛ t❡st ❢✉♥❝t✐♦♥ f ✐♥ t❤❡ ✈❡❝t♦r✲✈❛❧✉❡❞ ✈❛r✐❛t✐♦♥❛❧ ❢♦r♠✉❧❛t✐♦♥✳ ▲❡t pv = p(Cv) ❛♥❞ rv = r(Cv) ❢♦r v ∈ V ✳ ❲❡ ❝♦♥s✐❞❡r

✷ ✸ ❣✐✈❡♥ ❜②

❢♦r ✳ ❚❤❡♥

✷

✷ ✸

✷

.

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

Pr♦♦❢ ♦❢ t❤❡ ▲❡♠♠❛

■❞❡❛

❯s❡ ❛ ❝✐r❝❧❡✲♣❛❝❦✐♥❣ (Cv)v∈V r❡♣r❡s❡♥t✐♥❣ t❤❡ ♣❧❛♥❛r ❣r❛♣❤ G t♦ ♦❜t❛✐♥ ❛ t❡st ❢✉♥❝t✐♦♥ f ✐♥ t❤❡ ✈❡❝t♦r✲✈❛❧✉❡❞ ✈❛r✐❛t✐♦♥❛❧ ❢♦r♠✉❧❛t✐♦♥✳ ▲❡t pv = p(Cv) ❛♥❞ rv = r(Cv) ❢♦r v ∈ V ✳ ❲❡ ❝♦♥s✐❞❡r f ∈ l✷

m(V ; C✸) ❣✐✈❡♥ ❜② f (v) = pv ❢♦r v ∈ V ✳

❚❤❡♥

✷

✷ ✸

✷

.

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

Pr♦♦❢ ♦❢ t❤❡ ▲❡♠♠❛

■❞❡❛

❯s❡ ❛ ❝✐r❝❧❡✲♣❛❝❦✐♥❣ (Cv)v∈V r❡♣r❡s❡♥t✐♥❣ t❤❡ ♣❧❛♥❛r ❣r❛♣❤ G t♦ ♦❜t❛✐♥ ❛ t❡st ❢✉♥❝t✐♦♥ f ✐♥ t❤❡ ✈❡❝t♦r✲✈❛❧✉❡❞ ✈❛r✐❛t✐♦♥❛❧ ❢♦r♠✉❧❛t✐♦♥✳ ▲❡t pv = p(Cv) ❛♥❞ rv = r(Cv) ❢♦r v ∈ V ✳ ❲❡ ❝♦♥s✐❞❡r f ∈ l✷

m(V ; C✸) ❣✐✈❡♥ ❜② f (v) = pv ❢♦r v ∈ V ✳

❚❤❡♥ ||f ||✷

l✷

m(V ;C✸) =

• v∈V

m(v)|pv|✷ .

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

Pr♦♦❢ ♦❢ t❤❡ ▲❡♠♠❛

■❞❡❛

❯s❡ ❛ ❝✐r❝❧❡✲♣❛❝❦✐♥❣ (Cv)v∈V r❡♣r❡s❡♥t✐♥❣ t❤❡ ♣❧❛♥❛r ❣r❛♣❤ G t♦ ♦❜t❛✐♥ ❛ t❡st ❢✉♥❝t✐♦♥ f ✐♥ t❤❡ ✈❡❝t♦r✲✈❛❧✉❡❞ ✈❛r✐❛t✐♦♥❛❧ ❢♦r♠✉❧❛t✐♦♥✳ ▲❡t pv = p(Cv) ❛♥❞ rv = r(Cv) ❢♦r v ∈ V ✳ ❲❡ ❝♦♥s✐❞❡r f ∈ l✷

m(V ; C✸) ❣✐✈❡♥ ❜② f (v) = pv ❢♦r v ∈ V ✳

❚❤❡♥ ||f ||✷

l✷

m(V ;C✸) =

• v∈V

m(v)|pv|✷ = m(V ).

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

Pr♦♦❢ ♦❢ t❤❡ ▲❡♠♠❛

▼♦r❡♦✈❡r q(f ) =

• e={u,v}∈E

µ(e)|pu − pv|✷ ✷

✷ ✷

✷

✷

✷

✷

✽ ✭✷✮

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

Pr♦♦❢ ♦❢ t❤❡ ▲❡♠♠❛

▼♦r❡♦✈❡r q(f ) =

• e={u,v}∈E

µ(e)|pu − pv|✷ ≤ ✷

• e={u,v}∈E

µ(e)(r✷

u + r✷ v )

✷

✷

✷

✷

✽ ✭✷✮

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

Pr♦♦❢ ♦❢ t❤❡ ▲❡♠♠❛

▼♦r❡♦✈❡r q(f ) =

• e={u,v}∈E

µ(e)|pu − pv|✷ ≤ ✷

• e={u,v}∈E

µ(e)(r✷

u + r✷ v )

= ✷

• v∈V

dµ

v r✷ v

✷

✷

✽ ✭✷✮

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

Pr♦♦❢ ♦❢ t❤❡ ▲❡♠♠❛

▼♦r❡♦✈❡r q(f ) =

• e={u,v}∈E

µ(e)|pu − pv|✷ ≤ ✷

• e={u,v}∈E

µ(e)(r✷

u + r✷ v )

= ✷

• v∈V

dµ

v r✷ v

≤ ✷dµ

max

• v∈V

r✷

v .

✽ ✭✷✮

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

Pr♦♦❢ ♦❢ t❤❡ ▲❡♠♠❛

▼♦r❡♦✈❡r q(f ) =

• e={u,v}∈E

µ(e)|pu − pv|✷ ≤ ✷

• e={u,v}∈E

µ(e)(r✷

u + r✷ v )

= ✷

• v∈V

dµ

v r✷ v

≤ ✷dµ

max

• v∈V

r✷

v .

≤ ✽dµ

max

✭✷✮

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

Pr♦♦❢ ♦❢ t❤❡ ▲❡♠♠❛

❯s✐♥❣ t❤❡s❡ t✇♦ ❡st✐♠❛t❡s ✇❡ ♦❜t❛✐♥ q(f ) ||f ||l✷(V ;C✸) ≤ ✽ dµ

max

m(V ). ❇② ❛ss✉♠♣t✐♦♥ ✵ ❚❤❡r❡❢♦r❡ t❤❡ ♠✐♥✲♠❛① ♣r✐♥❝✐♣❧❡ ②✐❡❧❞s

✶

✽

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

Pr♦♦❢ ♦❢ t❤❡ ▲❡♠♠❛

❯s✐♥❣ t❤❡s❡ t✇♦ ❡st✐♠❛t❡s ✇❡ ♦❜t❛✐♥ q(f ) ||f ||l✷(V ;C✸) ≤ ✽ dµ

max

m(V ). ❇② ❛ss✉♠♣t✐♦♥

• v∈V

m(v)f (v) =

• v∈V

m(v)pv = ✵. ❚❤❡r❡❢♦r❡ t❤❡ ♠✐♥✲♠❛① ♣r✐♥❝✐♣❧❡ ②✐❡❧❞s

✶

✽

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

Pr♦♦❢ ♦❢ t❤❡ ▲❡♠♠❛

❯s✐♥❣ t❤❡s❡ t✇♦ ❡st✐♠❛t❡s ✇❡ ♦❜t❛✐♥ q(f ) ||f ||l✷(V ;C✸) ≤ ✽ dµ

max

m(V ). ❇② ❛ss✉♠♣t✐♦♥

• v∈V

m(v)f (v) =

• v∈V

m(v)pv = ✵. ❚❤❡r❡❢♦r❡ t❤❡ ♠✐♥✲♠❛① ♣r✐♥❝✐♣❧❡ ②✐❡❧❞s λ✶(L) ≤ ✽ dµ

max

m(V ).

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

❯♥✐q✉❡♥❡ss ♦❢ ❝✐r❝❧❡ ♣❛❝❦✐♥❣s

◗✉❡st✐♦♥

❍♦✇ ❞♦ ✇❡ ✜♥❞ ❛ ❝✐r❝❧❡✲♣❛❝❦✐♥❣✱ s✉❝❤ t❤❛t t❤❡ ♦rt❤♦❣♦♥❛❧✐t② ❝♦♥❞✐t✐♦♥ ❛❜♦✈❡ ✐s ❢✉❧✜❧❧❡❞❄

❚❤❡♦r❡♠ ✭❑♦❡❜❡ ✬✸✻✱ ❆♥❞r❡❡✈ ✬✼✵✱ ❚❤✉rst♦♥ ✬✼✽✮

■❢ ✐s ♠❛①✐♠❛❧ ♣❧❛♥❛r✱ t❤❡♥ t❤❡ ❝✐r❝❧❡ ♣❛❝❦✐♥❣ r❡♣r❡s❡♥t❛t✐♦♥ ✐♥

✷ ✐s

✉♥✐q✉❡ ✉♣ t♦ ❝♦♥❢♦r♠❛❧ ♠❛♣s ❛♥❞ r❡✢❡❝t✐♦♥s ✐♥

✷✳

❈❛♥♦♥✐❝❛❧ ❆♣♣r♦❛❝❤

❋✐♥❞ ❛ ❝♦♥❢♦r♠❛❧ ♠❛♣

✷ ✷ t❤❛t ♠❛♣s ❛ ❣✐✈❡♥ ❝✐r❝❧❡ ♣❛❝❦✐♥❣ t♦ ❛ ❝✐r❝❧❡

♣❛❝❦✐♥❣ t❤❛t s❛t✐s✜❡s t❤❡ ♦rt❤♦❣♦♥❛❧✐t② ❝♦♥❞✐t✐♦♥✳

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

❯♥✐q✉❡♥❡ss ♦❢ ❝✐r❝❧❡ ♣❛❝❦✐♥❣s

◗✉❡st✐♦♥

❍♦✇ ❞♦ ✇❡ ✜♥❞ ❛ ❝✐r❝❧❡✲♣❛❝❦✐♥❣✱ s✉❝❤ t❤❛t t❤❡ ♦rt❤♦❣♦♥❛❧✐t② ❝♦♥❞✐t✐♦♥ ❛❜♦✈❡ ✐s ❢✉❧✜❧❧❡❞❄

❚❤❡♦r❡♠ ✭❑♦❡❜❡ ✬✸✻✱ ❆♥❞r❡❡✈ ✬✼✵✱ ❚❤✉rst♦♥ ✬✼✽✮

■❢ G ✐s ♠❛①✐♠❛❧ ♣❧❛♥❛r✱ t❤❡♥ t❤❡ ❝✐r❝❧❡ ♣❛❝❦✐♥❣ r❡♣r❡s❡♥t❛t✐♦♥ ✐♥ S✷ ✐s ✉♥✐q✉❡ ✉♣ t♦ ❝♦♥❢♦r♠❛❧ ♠❛♣s ❛♥❞ r❡✢❡❝t✐♦♥s ✐♥ S✷✳

❈❛♥♦♥✐❝❛❧ ❆♣♣r♦❛❝❤

❋✐♥❞ ❛ ❝♦♥❢♦r♠❛❧ ♠❛♣

✷ ✷ t❤❛t ♠❛♣s ❛ ❣✐✈❡♥ ❝✐r❝❧❡ ♣❛❝❦✐♥❣ t♦ ❛ ❝✐r❝❧❡

♣❛❝❦✐♥❣ t❤❛t s❛t✐s✜❡s t❤❡ ♦rt❤♦❣♦♥❛❧✐t② ❝♦♥❞✐t✐♦♥✳

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

❯♥✐q✉❡♥❡ss ♦❢ ❝✐r❝❧❡ ♣❛❝❦✐♥❣s

◗✉❡st✐♦♥

❍♦✇ ❞♦ ✇❡ ✜♥❞ ❛ ❝✐r❝❧❡✲♣❛❝❦✐♥❣✱ s✉❝❤ t❤❛t t❤❡ ♦rt❤♦❣♦♥❛❧✐t② ❝♦♥❞✐t✐♦♥ ❛❜♦✈❡ ✐s ❢✉❧✜❧❧❡❞❄

❚❤❡♦r❡♠ ✭❑♦❡❜❡ ✬✸✻✱ ❆♥❞r❡❡✈ ✬✼✵✱ ❚❤✉rst♦♥ ✬✼✽✮

■❢ G ✐s ♠❛①✐♠❛❧ ♣❧❛♥❛r✱ t❤❡♥ t❤❡ ❝✐r❝❧❡ ♣❛❝❦✐♥❣ r❡♣r❡s❡♥t❛t✐♦♥ ✐♥ S✷ ✐s ✉♥✐q✉❡ ✉♣ t♦ ❝♦♥❢♦r♠❛❧ ♠❛♣s ❛♥❞ r❡✢❡❝t✐♦♥s ✐♥ S✷✳

❈❛♥♦♥✐❝❛❧ ❆♣♣r♦❛❝❤

❋✐♥❞ ❛ ❝♦♥❢♦r♠❛❧ ♠❛♣ S✷ → S✷ t❤❛t ♠❛♣s ❛ ❣✐✈❡♥ ❝✐r❝❧❡ ♣❛❝❦✐♥❣ t♦ ❛ ❝✐r❝❧❡ ♣❛❝❦✐♥❣ t❤❛t s❛t✐s✜❡s t❤❡ ♦rt❤♦❣♦♥❛❧✐t② ❝♦♥❞✐t✐♦♥✳

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

❆ ❯♥✐❢♦r♠✐③❛t✐♦♥ ❚❤❡♦r❡♠

❚❤❡♦r❡♠ ✭P✳ ✬✶✾✮

▲❡t G ❜❡ ♣❧❛♥❛r ❛♥❞ (Cv)v∈V ❜❡ ❛ ❝✐r❝❧❡ ♣❛❝❦✐♥❣ ❢♦r G ✐♥ S✷✳ ❆ss✉♠❡ t❤❛t m(V ) > ✷(m(u) + m(v)) ❢♦r ❛❧❧ {u, v} ∈ E, t❤❡♥ t❤❡r❡ ❡①✐sts ❛ ❝♦♥❢♦r♠❛❧ ♠❛♣

✷ ✷ s✉❝❤ t❤❛t

✵

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

❆ ❯♥✐❢♦r♠✐③❛t✐♦♥ ❚❤❡♦r❡♠

❚❤❡♦r❡♠ ✭P✳ ✬✶✾✮

▲❡t G ❜❡ ♣❧❛♥❛r ❛♥❞ (Cv)v∈V ❜❡ ❛ ❝✐r❝❧❡ ♣❛❝❦✐♥❣ ❢♦r G ✐♥ S✷✳ ❆ss✉♠❡ t❤❛t m(V ) > ✷(m(u) + m(v)) ❢♦r ❛❧❧ {u, v} ∈ E, t❤❡♥ t❤❡r❡ ❡①✐sts ❛ ❝♦♥❢♦r♠❛❧ ♠❛♣ f : S✷ → S✷, s✉❝❤ t❤❛t

• v∈V

m(v)p

• f (Cv)
• = ✵.

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

■❞❡❛ ♦❢ t❤❡ ♣r♦♦❢

❆❢t❡r r❡s❝❛❧✐♥❣ ✇❡ ♠❛② ❛ss✉♠❡ m(V ) = ✶✳ ❈♦♥s✐❞❡r ❛ ❝❡rt❛✐♥ ❢❛♠✐❧② ♦❢ ❝♦♥❢♦r♠❛❧ ♠❛♣s

✷ ✷ ❢♦r ✸

✶✱ t❤❛t ♠♦✈❡s t❤❡ ❝✐r❝❧❡s ❛❧♦♥❣ t❤❡ s♣❤❡r❡ ✐♥ ❛ ❝♦♥✈❡♥✐❡♥t ✇❛②✳ ❉❡✜♥❡ t❤❡ ♠❛♣ ❣✐✈❡♥ ❜② ❯s✐♥❣ ❛ ✜①♣♦✐♥t ❛r❣✉♠❡♥t ♦♥❡ ♠❛② s❤♦✇ t❤❛t ♠✉st ❜❡ ✵ ❢♦r s♦♠❡ ✉♥❞❡r t❤❡ ❛ss✉♠♣t✐♦♥ ♦♥ t❤❡ ✈❡rt❡① ✇❡✐❣❤t ✳

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

■❞❡❛ ♦❢ t❤❡ ♣r♦♦❢

❆❢t❡r r❡s❝❛❧✐♥❣ ✇❡ ♠❛② ❛ss✉♠❡ m(V ) = ✶✳ ❈♦♥s✐❞❡r ❛ ❝❡rt❛✐♥ ❢❛♠✐❧② ♦❢ ❝♦♥❢♦r♠❛❧ ♠❛♣s fα : S✷ → S✷ ❢♦r α ∈ R✸, |α| < ✶✱ t❤❛t ♠♦✈❡s t❤❡ ❝✐r❝❧❡s ❛❧♦♥❣ t❤❡ s♣❤❡r❡ ✐♥ ❛ ❝♦♥✈❡♥✐❡♥t ✇❛②✳ ❉❡✜♥❡ t❤❡ ♠❛♣ ❣✐✈❡♥ ❜② ❯s✐♥❣ ❛ ✜①♣♦✐♥t ❛r❣✉♠❡♥t ♦♥❡ ♠❛② s❤♦✇ t❤❛t ♠✉st ❜❡ ✵ ❢♦r s♦♠❡ ✉♥❞❡r t❤❡ ❛ss✉♠♣t✐♦♥ ♦♥ t❤❡ ✈❡rt❡① ✇❡✐❣❤t ✳

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

■❞❡❛ ♦❢ t❤❡ ♣r♦♦❢

❆❢t❡r r❡s❝❛❧✐♥❣ ✇❡ ♠❛② ❛ss✉♠❡ m(V ) = ✶✳ ❈♦♥s✐❞❡r ❛ ❝❡rt❛✐♥ ❢❛♠✐❧② ♦❢ ❝♦♥❢♦r♠❛❧ ♠❛♣s fα : S✷ → S✷ ❢♦r α ∈ R✸, |α| < ✶✱ t❤❛t ♠♦✈❡s t❤❡ ❝✐r❝❧❡s ❛❧♦♥❣ t❤❡ s♣❤❡r❡ ✐♥ ❛ ❝♦♥✈❡♥✐❡♥t ✇❛②✳ ❉❡✜♥❡ t❤❡ ♠❛♣ Φ ❣✐✈❡♥ ❜② Φ(α) =

• v∈V

m(v)p (fα(Cv)) . ❯s✐♥❣ ❛ ✜①♣♦✐♥t ❛r❣✉♠❡♥t ♦♥❡ ♠❛② s❤♦✇ t❤❛t ♠✉st ❜❡ ✵ ❢♦r s♦♠❡ ✉♥❞❡r t❤❡ ❛ss✉♠♣t✐♦♥ ♦♥ t❤❡ ✈❡rt❡① ✇❡✐❣❤t ✳

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

■❞❡❛ ♦❢ t❤❡ ♣r♦♦❢

❆❢t❡r r❡s❝❛❧✐♥❣ ✇❡ ♠❛② ❛ss✉♠❡ m(V ) = ✶✳ ❈♦♥s✐❞❡r ❛ ❝❡rt❛✐♥ ❢❛♠✐❧② ♦❢ ❝♦♥❢♦r♠❛❧ ♠❛♣s fα : S✷ → S✷ ❢♦r α ∈ R✸, |α| < ✶✱ t❤❛t ♠♦✈❡s t❤❡ ❝✐r❝❧❡s ❛❧♦♥❣ t❤❡ s♣❤❡r❡ ✐♥ ❛ ❝♦♥✈❡♥✐❡♥t ✇❛②✳ ❉❡✜♥❡ t❤❡ ♠❛♣ Φ ❣✐✈❡♥ ❜② Φ(α) =

• v∈V

m(v)p (fα(Cv)) . ❯s✐♥❣ ❛ ✜①♣♦✐♥t ❛r❣✉♠❡♥t ♦♥❡ ♠❛② s❤♦✇ t❤❛t Φ ♠✉st ❜❡ ✵ ❢♦r s♦♠❡ α ✉♥❞❡r t❤❡ ❛ss✉♠♣t✐♦♥ ♦♥ t❤❡ ✈❡rt❡① ✇❡✐❣❤t m✳

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

❆♥ ❡✐❣❡♥✈❛❧✉❡ ❜♦✉♥❞ ❢♦r ♣❧❛♥❛r ❝♦♠❜✐♥❛t♦r✐❛❧ ❣r❛♣❤s

❈♦r♦❧❧❛r②

▲❡t G ❜❡ ♣❧❛♥❛r ❛♥❞ ❛ss✉♠❡ t❤❛t m(V ) > ✷(m(v) + m(u)) ❢♦r ❛❧❧ {u, v} ∈ E, t❤❡♥ ✇❡ ♦❜t❛✐♥ t❤❡ ❡st✐♠❛t❡ λ✶(L) ≤ ✽ dµ

max

m(V ).

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

❆♥ ❡✐❣❡♥✈❛❧✉❡ ❜♦✉♥❞ ❢♦r ♣❧❛♥❛r q✉❛♥t✉♠ ❣r❛♣❤s

❈♦r♦❧❧❛r②

▲❡t G ❜❡ ♣❧❛♥❛r ❛♥❞ ❛ss✉♠❡ t❤❛t L > dl

u + dl v ❢♦r ❛❧❧ {u, v} ∈ E,

t❤❡♥ ✇❡ ♦❜t❛✐♥ t❤❡ ❡st✐♠❛t❡ λ✶(−∆) ≤ ✷✹dµ

max

L , ✇❤❡r❡ µ(e) = ✶

le ✳

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

❍♦✇ t♦ ❞r♦♣ t❤❡ ❝♦♥❞✐t✐♦♥ L > dl

v + dl u❄

❈♦♥s✐❞❡r ❛♥ ❛r❜✐tr❛r② ♠❡tr✐❝ ❣r❛♣❤ G = (G, lG) ♦✈❡r s♦♠❡ ❝♦♠❜✐♥❛t♦r✐❛❧✱ ❝♦♥♥❡❝t❡❞ ❛♥❞ ✜♥✐t❡ ❣r❛♣❤ G = (V (G), E(G))✱ t❤❛t ✐s ♥♦t ♥❡❝❡ss❛r✐❧② s✐♠♣❧❡✳ ▲❡t ❜❡ t❤❡ s✉❜❞✐✈s✐♦♥ ❣r❛♣❤ ♦❜t❛✐♥❡❞ ❛❢t❡r ❞✐✈✐❞✐♥❣ ❡❛❝❤ ❡❞❣❡ ✐♥t♦ ❢♦✉r ❡❞❣❡s ♦❢ ❡q✉❛❧ ❧❡♥❣t❤✳ ❲❡ s❤❛❧❧ ✇r✐t❡ ◆♦t❡

✶ ✶

❋✐❣✉r❡✿ ❙✉❜❞✐✈s✐♦♥ ♦❢

✹

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

❍♦✇ t♦ ❞r♦♣ t❤❡ ❝♦♥❞✐t✐♦♥ L > dl

v + dl u❄

❈♦♥s✐❞❡r ❛♥ ❛r❜✐tr❛r② ♠❡tr✐❝ ❣r❛♣❤ G = (G, lG) ♦✈❡r s♦♠❡ ❝♦♠❜✐♥❛t♦r✐❛❧✱ ❝♦♥♥❡❝t❡❞ ❛♥❞ ✜♥✐t❡ ❣r❛♣❤ G = (V (G), E(G))✱ t❤❛t ✐s ♥♦t ♥❡❝❡ss❛r✐❧② s✐♠♣❧❡✳ ▲❡t G′ = (G ′, l′) ❜❡ t❤❡ s✉❜❞✐✈s✐♦♥ ❣r❛♣❤ ♦❜t❛✐♥❡❞ ❛❢t❡r ❞✐✈✐❞✐♥❣ ❡❛❝❤ ❡❞❣❡ ✐♥t♦ ❢♦✉r ❡❞❣❡s ♦❢ ❡q✉❛❧ ❧❡♥❣t❤✳ ❲❡ s❤❛❧❧ ✇r✐t❡ V (G ′) = Vnew ∪ Vold. ◆♦t❡

✶ ✶

❋✐❣✉r❡✿ ❙✉❜❞✐✈s✐♦♥ ♦❢

✹

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

❍♦✇ t♦ ❞r♦♣ t❤❡ ❝♦♥❞✐t✐♦♥ L > dl

v + dl u❄

❈♦♥s✐❞❡r ❛♥ ❛r❜✐tr❛r② ♠❡tr✐❝ ❣r❛♣❤ G = (G, lG) ♦✈❡r s♦♠❡ ❝♦♠❜✐♥❛t♦r✐❛❧✱ ❝♦♥♥❡❝t❡❞ ❛♥❞ ✜♥✐t❡ ❣r❛♣❤ G = (V (G), E(G))✱ t❤❛t ✐s ♥♦t ♥❡❝❡ss❛r✐❧② s✐♠♣❧❡✳ ▲❡t G′ = (G ′, l′) ❜❡ t❤❡ s✉❜❞✐✈s✐♦♥ ❣r❛♣❤ ♦❜t❛✐♥❡❞ ❛❢t❡r ❞✐✈✐❞✐♥❣ ❡❛❝❤ ❡❞❣❡ ✐♥t♦ ❢♦✉r ❡❞❣❡s ♦❢ ❡q✉❛❧ ❧❡♥❣t❤✳ ❲❡ s❤❛❧❧ ✇r✐t❡ V (G ′) = Vnew ∪ Vold. ◆♦t❡

✶ ✶

❋✐❣✉r❡✿ ❙✉❜❞✐✈s✐♦♥ ♦❢

✹

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

❍♦✇ t♦ ❞r♦♣ t❤❡ ❝♦♥❞✐t✐♦♥ L > dl

v + dl u❄

❈♦♥s✐❞❡r ❛♥ ❛r❜✐tr❛r② ♠❡tr✐❝ ❣r❛♣❤ G = (G, lG) ♦✈❡r s♦♠❡ ❝♦♠❜✐♥❛t♦r✐❛❧✱ ❝♦♥♥❡❝t❡❞ ❛♥❞ ✜♥✐t❡ ❣r❛♣❤ G = (V (G), E(G))✱ t❤❛t ✐s ♥♦t ♥❡❝❡ss❛r✐❧② s✐♠♣❧❡✳ ▲❡t G′ = (G ′, l′) ❜❡ t❤❡ s✉❜❞✐✈s✐♦♥ ❣r❛♣❤ ♦❜t❛✐♥❡❞ ❛❢t❡r ❞✐✈✐❞✐♥❣ ❡❛❝❤ ❡❞❣❡ ✐♥t♦ ❢♦✉r ❡❞❣❡s ♦❢ ❡q✉❛❧ ❧❡♥❣t❤✳ ❲❡ s❤❛❧❧ ✇r✐t❡ V (G ′) = Vnew ∪ Vold. ◆♦t❡

✶ ✶

❋✐❣✉r❡✿ ❙✉❜❞✐✈s✐♦♥ ♦❢ K✹

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

❍♦✇ t♦ ❞r♦♣ t❤❡ ❝♦♥❞✐t✐♦♥ L > dl

v + dl u❄

❈♦♥s✐❞❡r ❛♥ ❛r❜✐tr❛r② ♠❡tr✐❝ ❣r❛♣❤ G = (G, lG) ♦✈❡r s♦♠❡ ❝♦♠❜✐♥❛t♦r✐❛❧✱ ❝♦♥♥❡❝t❡❞ ❛♥❞ ✜♥✐t❡ ❣r❛♣❤ G = (V (G), E(G))✱ t❤❛t ✐s ♥♦t ♥❡❝❡ss❛r✐❧② s✐♠♣❧❡✳ ▲❡t G′ = (G ′, l′) ❜❡ t❤❡ s✉❜❞✐✈s✐♦♥ ❣r❛♣❤ ♦❜t❛✐♥❡❞ ❛❢t❡r ❞✐✈✐❞✐♥❣ ❡❛❝❤ ❡❞❣❡ ✐♥t♦ ❢♦✉r ❡❞❣❡s ♦❢ ❡q✉❛❧ ❧❡♥❣t❤✳ ❲❡ s❤❛❧❧ ✇r✐t❡ V (G ′) = Vnew ∪ Vold. ◆♦t❡

✶ ✶

❋✐❣✉r❡✿ ❙✉❜❞✐✈s✐♦♥ ♦❢ K✹

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

❍♦✇ t♦ ❞r♦♣ t❤❡ ❝♦♥❞✐t✐♦♥ L > dl

v + dl u❄

❈♦♥s✐❞❡r ❛♥ ❛r❜✐tr❛r② ♠❡tr✐❝ ❣r❛♣❤ G = (G, lG) ♦✈❡r s♦♠❡ ❝♦♠❜✐♥❛t♦r✐❛❧✱ ❝♦♥♥❡❝t❡❞ ❛♥❞ ✜♥✐t❡ ❣r❛♣❤ G = (V (G), E(G))✱ t❤❛t ✐s ♥♦t ♥❡❝❡ss❛r✐❧② s✐♠♣❧❡✳ ▲❡t G′ = (G ′, l′) ❜❡ t❤❡ s✉❜❞✐✈s✐♦♥ ❣r❛♣❤ ♦❜t❛✐♥❡❞ ❛❢t❡r ❞✐✈✐❞✐♥❣ ❡❛❝❤ ❡❞❣❡ ✐♥t♦ ❢♦✉r ❡❞❣❡s ♦❢ ❡q✉❛❧ ❧❡♥❣t❤✳ ❲❡ s❤❛❧❧ ✇r✐t❡ V (G ′) = Vnew ∪ Vold. ◆♦t❡ L′ = L, λ✶(−∆G′) = λ✶(−∆G).

❋✐❣✉r❡✿ ❙✉❜❞✐✈s✐♦♥ ♦❢ K✹

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

❍♦✇ t♦ ❞r♦♣ t❤❡ ❝♦♥❞✐t✐♦♥ L > dl

v + dl u❄

❲❡ ❤❛✈❡ dl′

v =

• ✹

✷

❛♥❞ ✐s ♦♥ ❛♥❞ t❤✉s ❤♦❧❞s ❢♦r ✇✐t❤ ✱ s♦ ❜② ♦✉r ❧❛st ❝♦r♦❧❧❛r②

✶

✷✹ ✶✾✷ .

❋✐❣✉r❡✿ ❙✉❜❞✐✈s✐♦♥ ♦❢ K✹

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

❍♦✇ t♦ ❞r♦♣ t❤❡ ❝♦♥❞✐t✐♦♥ L > dl

v + dl u❄

❲❡ ❤❛✈❡ dl′

v =

dl

v

✹ ,

v ∈ Vold,

✷

❛♥❞ ✐s ♦♥ ❛♥❞ t❤✉s ❤♦❧❞s ❢♦r ✇✐t❤ ✱ s♦ ❜② ♦✉r ❧❛st ❝♦r♦❧❧❛r②

✶

✷✹ ✶✾✷ .

❋✐❣✉r❡✿ ❙✉❜❞✐✈s✐♦♥ ♦❢ K✹

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

❍♦✇ t♦ ❞r♦♣ t❤❡ ❝♦♥❞✐t✐♦♥ L > dl

v + dl u❄

❲❡ ❤❛✈❡ dl′

v =

dl

v

✹ ,

v ∈ Vold,

le ✷ ,

v ∈ Vnew ❛♥❞ v ✐s ♦♥ e. ❛♥❞ t❤✉s ❤♦❧❞s ❢♦r ✇✐t❤ ✱ s♦ ❜② ♦✉r ❧❛st ❝♦r♦❧❧❛r②

✶

✷✹ ✶✾✷ .

❋✐❣✉r❡✿ ❙✉❜❞✐✈s✐♦♥ ♦❢ K✹

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

❍♦✇ t♦ ❞r♦♣ t❤❡ ❝♦♥❞✐t✐♦♥ L > dl

v + dl u❄

❲❡ ❤❛✈❡ dl′

v =

dl

v

✹ ,

v ∈ Vold,

le ✷ ,

v ∈ Vnew ❛♥❞ v ✐s ♦♥ e. ❛♥❞ t❤✉s dl′

v + dl′ u < L

❤♦❧❞s ❢♦r u, v ∈ V (G ′) ✇✐t❤ u ∼ v ✱ s♦ ❜② ♦✉r ❧❛st ❝♦r♦❧❧❛r②

✶

✷✹ ✶✾✷ .

❋✐❣✉r❡✿ ❙✉❜❞✐✈s✐♦♥ ♦❢ K✹

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

❍♦✇ t♦ ❞r♦♣ t❤❡ ❝♦♥❞✐t✐♦♥ L > dl

v + dl u❄

❲❡ ❤❛✈❡ dl′

v =

dl

v

✹ ,

v ∈ Vold,

le ✷ ,

v ∈ Vnew ❛♥❞ v ✐s ♦♥ e. ❛♥❞ t❤✉s dl′

v + dl′ u < L

❤♦❧❞s ❢♦r u, v ∈ V (G ′) ✇✐t❤ u ∼ v✱ s♦ ❜② ♦✉r ❧❛st ❝♦r♦❧❧❛r② λ✶(−∆) ≤ ✷✹dµ′

max

L ✶✾✷ .

❋✐❣✉r❡✿ ❙✉❜❞✐✈s✐♦♥ ♦❢ K✹

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

❍♦✇ t♦ ❞r♦♣ t❤❡ ❝♦♥❞✐t✐♦♥ L > dl

v + dl u❄

❲❡ ❤❛✈❡ dl′

v =

dl

v

✹ ,

v ∈ Vold,

le ✷ ,

v ∈ Vnew ❛♥❞ v ✐s ♦♥ e. ❛♥❞ t❤✉s dl′

v + dl′ u < L

❤♦❧❞s ❢♦r u, v ∈ V (G ′) ✇✐t❤ u ∼ v✱ s♦ ❜② ♦✉r ❧❛st ❝♦r♦❧❧❛r② λ✶(−∆) ≤ ✷✹dµ′

max

L ≤ ✶✾✷dµ

max

L .

❋✐❣✉r❡✿ ❙✉❜❞✐✈s✐♦♥ ♦❢ K✹

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

❙✉♠♠❛r②

❚❤❡♦r❡♠ ✭P✳✬✶✾✮

■❢ G = (G, l) ✐s ❛ ♣❧❛♥❛r✱ ✜♥✐t❡✱ ❝♦♠♣❛❝t ❛♥❞ ❝♦♥♥❡❝t❡❞ ♠❡tr✐❝ ❣r❛♣❤✱ t❤❡♥ ✇❡ ❤❛✈❡ t❤❡ s♣❡❝tr❛❧ ❜♦✉♥❞ λ✶(−∆) ≤ ✶✾✷dµ

max

L .

❘❡♠❛r❦

❚❤❡ ♣❧❛♥❛r✐t② ❛ss✉♠♣t✐♦♥ ❝❛♥♥♦t ❜❡ ❞r♦♣♣❡❞✦ ✭❊①❛♠♣❧❡✿ ❈♦♠♣❧❡t❡ ❡q✉✐❧❛t❡r❛❧ ❣r❛♣❤s ♦❢ ❝♦♥st❛♥t ❧❡♥❣t❤ ✶✳✮

❚♦ ❉♦

• ❡♥❡r❛❧✐③❡ ♦✉r r❡s✉❧ts t♦ ❤✐❣❤❡r ♦r❞❡r ❡✐❣❡♥✈❛❧✉❡s ❛♥❞ ❣r❛♣❤s ♦❢ ❤✐❣❤❡r

❣❡♥✉s✳

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

❙✉♠♠❛r②

❚❤❡♦r❡♠ ✭P✳✬✶✾✮

■❢ G = (G, l) ✐s ❛ ♣❧❛♥❛r✱ ✜♥✐t❡✱ ❝♦♠♣❛❝t ❛♥❞ ❝♦♥♥❡❝t❡❞ ♠❡tr✐❝ ❣r❛♣❤✱ t❤❡♥ ✇❡ ❤❛✈❡ t❤❡ s♣❡❝tr❛❧ ❜♦✉♥❞ λ✶(−∆) ≤ ✶✾✷dµ

max

L .

❘❡♠❛r❦

❚❤❡ ♣❧❛♥❛r✐t② ❛ss✉♠♣t✐♦♥ ❝❛♥♥♦t ❜❡ ❞r♦♣♣❡❞✦ ✭❊①❛♠♣❧❡✿ ❈♦♠♣❧❡t❡ ❡q✉✐❧❛t❡r❛❧ ❣r❛♣❤s ♦❢ ❝♦♥st❛♥t ❧❡♥❣t❤ l ≡ ✶✳✮

❚♦ ❉♦

• ❡♥❡r❛❧✐③❡ ♦✉r r❡s✉❧ts t♦ ❤✐❣❤❡r ♦r❞❡r ❡✐❣❡♥✈❛❧✉❡s ❛♥❞ ❣r❛♣❤s ♦❢ ❤✐❣❤❡r

❣❡♥✉s✳

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

❙✉♠♠❛r②

❚❤❡♦r❡♠ ✭P✳✬✶✾✮

■❢ G = (G, l) ✐s ❛ ♣❧❛♥❛r✱ ✜♥✐t❡✱ ❝♦♠♣❛❝t ❛♥❞ ❝♦♥♥❡❝t❡❞ ♠❡tr✐❝ ❣r❛♣❤✱ t❤❡♥ ✇❡ ❤❛✈❡ t❤❡ s♣❡❝tr❛❧ ❜♦✉♥❞ λ✶(−∆) ≤ ✶✾✷dµ

max

L .

❘❡♠❛r❦

❚❤❡ ♣❧❛♥❛r✐t② ❛ss✉♠♣t✐♦♥ ❝❛♥♥♦t ❜❡ ❞r♦♣♣❡❞✦ ✭❊①❛♠♣❧❡✿ ❈♦♠♣❧❡t❡ ❡q✉✐❧❛t❡r❛❧ ❣r❛♣❤s ♦❢ ❝♦♥st❛♥t ❧❡♥❣t❤ l ≡ ✶✳✮

❚♦ ❉♦

• ❡♥❡r❛❧✐③❡ ♦✉r r❡s✉❧ts t♦ ❤✐❣❤❡r ♦r❞❡r ❡✐❣❡♥✈❛❧✉❡s ❛♥❞ ❣r❛♣❤s ♦❢ ❤✐❣❤❡r

❣❡♥✉s✳

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

❚❤❛♥❦ ②♦✉ ❢♦r ②♦✉r ❛tt❡♥t✐♦♥✦

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