Préambule Notations

|X|X
G= (V, E)V E ⊂V×V
∀(x, y)∈E, x 6=y(y, x)∈E
V E
x y k ∈N
(xi)i=0,...,k x0=x xk=y(xi, xi+1)∈E i = 0, . . . , k −1
x, y ∈V k ∈Nx y
k G
x∈V x val(x)
{y∈V(x, y)∈E}G
x, y ∈Vval(x) = val(y)
G A B V V =A∪B
A∩B=∅E⊂A×B∪B×A
RVVR
hf, gi=Px∈Vf(x)g(x)f, g ∈RVk · k
TGRV
(TGf)(x) = X
y∈V(x,y)∈E
(f(x)−f(y)) x∈E
f∈RVqGqG(f) =
hTGf, f if∈RV
A
n λ16
· · · 6λnA

λ1, . . . , λnk∈ {1, . . . , n}
λkk A
A
n∈N∗Z/nZn
Mn(R)nGLn(R)
Mn(R)O(n) GLn(R)
End(V)
V
G= (V, E)n=|V|
n>2
TGRV
f∈RV
qG(f) = 1
2X
(x,y)∈E
(f(x)−f(y))2.
λ16· · · 6λn
TG
(e1, . . . , en)RVTG(ei) = λiei
i∈ {1,...n}
f∈RV
qG(f) =
n
X
i=1
λif2
i
(f1, . . . , fn)f(e1, . . . , en)
λ160
SRVk∈ {1,...n}Fk
<1+. . . <k−1RVk∈
{1, . . . , n}
λk= inf
f∈Fk∩SqG(f)
qG(f)>λkkfk2f∈Fk
k∈ {1, . . . , n} Wk
kRVk∈ {1, . . . , n}
λk= inf
W∈Wk sup
f∈W∩S
qG(f)!.

G
TG
λ1λ2>0
dG= max{val(x), x ∈V}
TG
2dGf TG
x∈V|f(x)|
G2dGTG
2dGTGG
G0= (V, E0)E0⊂E λ0
16· · · 6λ0
n
TG0
k∈ {1, . . . , n}λ0
k6λk
Kn({1, . . . , n},{(i, j)∈ {1, . . . , n}2i6=j})
TKn
G= (V, E)TG|V|
n∈N∗An{1, . . . , n}
+1
A3(123)
A4(123) (12)(34)
σ∈ A4σ(4) = 4
A4a= (12345) b= (12)(34)
G= (A5, E)
E={(g, h)∈ A4, g−1h∈ {a, a−1, b}}.
g∈ A5LgRV(Lgf)(x) =
f(g−1x)f∈RVx∈ A5
G
LgRV
∀g, h ∈ A5, Lg◦Lh=Lgh TG◦Lg=Lg◦TG.

TG
3A5
O(1) O(2)
ρ:A5→GL5(R)
ρ(g)(x) = xg−1(1), . . . , xg−1(5)
g∈ A5x= (x1, . . . , x5)∈R5
FR5x1+x2+x3+x4+x5= 0
g∈ A5F ρ(g)
˜ρ(g)∈End(F) ˜ρ(g)(x) = ρ(g)(x)
x∈F(˜ρ(g))g∈A5End(F)
λ∈RM∈End(F)
f∈RVf(g) = Tr(˜ρ(g)M)
TGλ
A=
3−2 0 0 −1
−2 3 −1 0 0
0−1 3 −2 0
0 0 −2 3 −1
−1 0 0 −1 2
P(x) = det(xI5−
A) = x(x−2)(x−5)(x2−7x+ 8) x∈R
G= (V, E)A V ∂A =
{(x, y)∈E x ∈A y ∈V\A}
hG= inf |∂A|
|A|, A ⊂V0<|A|6|V| 2.
x, y ∈V k ∈Nx y
k= 0 x=y d(x, y)
∆G= max
(x,y)∈V2d(x, y).
x, y, z ∈V d(x, y) = d(y, x)d(x, z)6
d(x, y) + d(y, z)

A V k ∈N
B(A, k) = {x∈V, ∃y∈A, d(x, y)6k}
dG= max{val(x), x ∈V}
B(A, 0) = A A V |A|6|V|/2
|B(A, 1)|>(1 + hG
dG
)|A|.
∆G62 ln(|V|/2)
ln(1 + hG/dG)+ 2.
λ2TG
A B V A∪B=V A∩B=∅
a=|A|b=|B|f:V→Rf|A=b
f|B=−a f TG
qG(f) = (a+b)|∂A|
ab kfk2.
λ262hG
λ2
f∈RVS(f) = 1
2P(x,y)∈E|f(x)2−f(y)2|
S(f)6p2dGqG(f)kfk
f:V→[0,+∞[|f−1(]0,+∞[)|6|V|/2
S(f)>hGkfk2V={x1, . . . , xn}
f(x1)>f(x2)· · · >f(xn)hG
A={x1, . . . , xk}k k 6n/2
λ26h2
G
2dG
E
k · k h·,·i E
Q:EV→RQ(f) = 1
2P(x,y)∈Ekf(x)−f(y)k2
f∈ EV
λ2= inf
f∈SQ(f)S={f:V→ E,X
x∈V
kf(x)k2= 1,X
x∈V
f(x) = 0}.
GEu:V→ E
D(x) = {w∈ E,kwk=
1,hu(x), w −u(x)i>0}x∈V