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EKSENTISITAS Misalkan πΊ adalah Graf terhubung dan π₯ adalah titik dari πΊ, Eksentisitas π(π₯) dari π₯ adalah nilai terbesar dari π·(π₯, π¦). Contoh 1 :
a
b
c
d
e
f
Graf G
Temukan π(π₯)untuk setiap titik pada πΊ. JAWAB: π(π) = π·(π, π) = π·(π, π) = 3 π(π) = π·(π, π) = π·(π, π) = 2 π(π) = π·(π, π) = π·(π, π) = 3 π(π) = π·(π, π) = 3 π(π) = π·(π, π) = 3 π(π) = π·(π, π) = π·(π, π) = 2
DIAMETER Diameter di graf πΊ adalah nilai maksimum dari eksentisitas titik-titik di πΊ. Simbol : π·(πΊ) =
ππππ π(π₯) π₯
Dari contoh 1 , diperoleh diameter π·(πΊ) = 3, yang merupakan nilai terbesar dari eksentrisitas graf G.
RADIUS Radius di graf πΊ adalah nilai terkecil dari eksentisitas titik-titik di πΊ. Simbol : π (πΊ) =
πππ π(π₯) π₯
Dari contoh 1 diperoleh Radius π (πΊ) = 2, yang merupakan nilai terkecil dari eksentrisitas graf G.
LATIHAN SOAL 1) Temukan Lilitan,Radius, dan Diameter graf dibawah. (i) πΏπ jawab : π(π₯) = {4,3,5,7,6,6,5,4,4,6,5,6,6,5,5} Ukuran :1 πππ Radius : π (πΊ) = π(π₯) = 3 π₯ ππππ Diameter : π·(πΊ) = π(π₯) = 7 π₯ (ii) Graf Petersen Jawab: π(π₯) = {1,2,3,2,3,2,3,1,2,1,2,3,2,1,3,2,1,1,1,2,4,1,3,2} Ukuran :3 πππ Radius : π (πΊ) = π(π₯) = 1 π₯ πππ₯ Diameter : π·(πΊ) = π(π₯) = 4 π₯ (iii)
Prisma Segilima Jawab: π(π₯) = {5,6,8,9} Ukuran :9 πππ π(π₯) = 5 π₯ πππ₯ : π·(πΊ) = π(π₯) = 9 π₯ : π (πΊ) =
Radius Diamater
2) Tentukan Lintasan, Keliling, Radius, dan Diameter dari graf berikut. i.
a
b
c
h
d g
Jawab: π·(π, π) = 1 π·(π, π) = 2 π·(π, π) = 2 π·(π, π) = 3 π·(π, π) = 2 π·(π, π) = 2 π·(π, β) = 1 π·(π, π) = 1 π·(π, π) = 2
f
e π·(π, π) = 1 π·(π, π) = 1 π·(π, π) = 2 π·(π, π) = 2 π·(π, π) = 2 π·(π, β) = 1
π·(π, π) = 1 π·(π, π) = 2 π·(π, π) = 2 π·(π, π) = 3 π·(π, β) = 2
π·(π, π) = 1 π·(π, β) = 1
π·(π, β) = 1
π·(π, π) = 1 π·(π, π) = 1 π·(π, π) = 1 π·(π, β) = 2
π·(π, β) = 2 Lilitan Keliling
: 3 (a-ab-b-bh-h-ha-a) :8 πππ : π (πΊ) = π(π₯) = 1 π₯ πππ₯ : π·(πΊ) = π(π₯) = 3 π₯
Radius Diameter ii.
a
b c
h
d
g f Jawab: π·(π, π) = 1 π·(π, π) = 2 π·(π, π) = 3 π·(π, π) = 4 π·(π, π) = 3 π·(π, π) = 2 π·(π, β) = 1 π·(π, π) = 1 π·(π, π) = 2 π·(π, β) = 3 Lilitan Keliling Radius Diameter
e π·(π, π) = 1 π·(π, π) = 2 π·(π, π) = 3 π·(π, π) = 1 π·(π, π) = 2 π·(π, β) = 2
π·(π, π) = 1 π·(π, π) = 2 π·(π, π) = 2 π·(π, π) = 3 π·(π, β) = 3
π·(π, π) = 1 π·(π, β) = 2
π·(π, β) = 1
: 5 (a-ab-b-bf-f-fg-g-gh-h-ha-a) :8 πππ : π (πΊ) = π(π₯) = 1 π₯ πππ₯ : π·(πΊ) = π(π₯) = 4 π₯
π·(π, π) = 1 π·(π, π) = 2 π·(π, π) = 3 π·(π, β) = 4
iii.
a
b c
h
d
g f
e
Jawab: π·(π, π) = 1 π·(π, π) = 2 π·(π, π) = 3 π·(π, π) = 4 π·(π, π) = 3 π·(π, π) = 2 π·(π, β) = 1 π·(π, π) = 1 π·(π, π) = 2 π·(π, β) = 3 Lilitan Keliling
π·(π, π) = 1 π·(π, π) = 2 π·(π, π) = 3 π·(π, π) = 4 π·(π, π) = 3 π·(π, β) = 2
π·(π, π) = 1 π·(π, π) = 2 π·(π, π) = 3 π·(π, π) = 2 π·(π, β) = 1
π·(π, π) = 1 π·(π, β) = 2
π·(π, β) = 1
π·(π, π) = 1 π·(π, π) = 2 π·(π, π) = 3 π·(π, β) = 2
: 4 (a-ab-b-bc-c-ch-h-ha-a) :8 πππ : π (πΊ) = π(π₯) = 1 π₯ πππ₯ : π·(πΊ) = π(π₯) = 4 π₯
Radius Diameter iv.
j
a
b
e
f
d
g i c
Jawab: π·(π, π) = 1 π·(π, π) = 2 π·(π, π) = 3 π·(π, π) = 4
h π·(π, π) = 1 π·(π, π) = 2 π·(π, π) = 3 π·(π, π) = 4
π·(π, π) = 1 π·(π, π) = 2 π·(π, π) = 3 π·(π, π) = 2
π·(π, π) = 1 π·(π, π) = 2 π·(π, π) = 1 π·(π, β) = 2
π·(π, π) = 5 π·(π, π) = 4 π·(π, β) = 3 π·(π, π) = 4 π·(π, π) = 5
π·(π, π) = 3 π·(π, β) = 2 π·(π, π) = 3 π·(π, π) = 4
π·(π, β) = 1 π·(π, π) = 2 π·(π, π) = 3
π·(π, π) = 3 π·(π, π) = 4
π·(π, π) = 1 π·(π, π) = 2 π·(π, β) = 3 π·(π, π) = 4 π·(π, π) = 5
π·(π, π) = 1 π·(π, β) = 2 π·(π, π) = 3 π·(π, π) = 4
π·(π, β) = 1 π·(π, π) = 2 π·(π, π) = 3
π·(β, π) = 1 π·(β, π) = 2
π·(π, π) = 1 Lilitan Keliling
: 6 (e-ef-f-fg-g-gh-h-hc-c-cd-d-de-d) :9 πππ : π (πΊ) = π(π₯) = 1 π₯ πππ₯ : π·(πΊ) = π(π₯) = 5 π₯
Radius Diameter v.
a
c
b g
j i
d f
e
h
Jawab: π·(π, π) = 2 π·(π, π) = 3 π·(π, π) = 4 π·(π, π) = 3 π·(π, π) = 3 π·(π, π) = 1 π·(π, β) = 2 π·(π, π) = 2 π·(π, π) = 3
π·(π, π) = 1 π·(π, π) = 2 π·(π, π) = 2 π·(π, π) = 1 π·(π, π) = 1 π·(π, β) = 2 π·(π, π) = 2 π·(π, π) = 3
π·(π, π) = 1 π·(π, π) = 1 π·(π, π) = 2 π·(π, π) = 2 π·(π, β) = 2 π·(π, π) = 3 π·(π, π) = 4
π·(π, π) = 2 π·(π, π) = 3 π·(π, π) = 3 π·(π, β) = 3 π·(π, π) = 4 π·(π, π) = 5
π·(π, π) = 2 π·(π, π) = 2 π·(π, β) = 1 π·(π, π) = 3
π·(π, π) = 2 π·(π, β) = 1 π·(π, π) = 3 π·(π, π) = 4
π·(π, β) = 1 π·(π, π) = 1 π·(π, π) = 2
π·(β, π) = 2 π·(β, π) = 3
π·(π, π) = 4 π·(π, π) = 1 Lilitan Keliling
: 4 (g-gb-b-bf-f-fh-h-hg-g) :5 πππ : π (πΊ) = π(π₯) = 1 π₯ πππ₯ : π·(πΊ) = π(π₯) = 5 π₯
Radius Diameter
3) Tentukan Lintasan, Keliling, radius dan diameter graf berikut. a.
a b c
f d
e
Jawab: π·(π, π) = 1 π·(π, π) = 2 π·(π, π) = 2 π·(π, π) = 2 π·(π, π) = 1
Lilitan Keliling
π·(π, π) = 1 π·(π, π) = 1 π·(π, π) = 2 π·(π, π) = 2
: 3 (a-ab-b-bf-f-fa-a) :6 πππ : π (πΊ) = π(π₯) = 1 π₯ πππ₯ : π·(πΊ) = π(π₯) = 2 π₯
Radius Diameter
b.
a
b c
h
d
g f
e
π·(π, π) = 1 π·(π, π) = 2 π·(π, π) = 2
π·(π, π) = 1 π·(π, π) = 1
Jawab: π·(π, π) = 1 π·(π, π) = 2 π·(π, π) = 3 π·(π, π) = 4 π·(π, π) = 3 π·(π, π) = 2 π·(π, β) = 1 π·(π, π) = 1 π·(π, π) = 2 π·(π, β) = 3 Lilitan Keliling
π·(π, π) = 1 π·(π, π) = 2 π·(π, π) = 3 π·(π, π) = 4 π·(π, π) = 3 π·(π, β) = 2
π·(π, π) = 1 π·(π, π) = 2 π·(π, π) = 3 π·(π, π) = 2 π·(π, β) = 1
π·(π, π) = 1 π·(π, β) = 2
π·(π, β) = 1
π·(π, π) = 1 π·(π, π) = 2 π·(π, π) = 1 π·(π, β) = 2
: 4 (a-ab-b-bc-c-ch-h-ha-a) :8 πππ : π (πΊ) = π(π₯) = 1 π₯ πππ₯ : π·(πΊ) = π(π₯) = 4 π₯
Radius Diameter c.
a
b c
h g
d f
Jawab: π·(π, π) = 1 π·(π, π) = 2 π·(π, π) = 3 π·(π, π) = 4 π·(π, π) = 3 π·(π, π) = 2 π·(π, β) = 1 π·(π, π) = 1 π·(π, π) = 2 π·(π, β) = 3 Lilitan Keliling Radius
e π·(π, π) = 1 π·(π, π) = 4 π·(π, π) = 5 π·(π, π) = 4 π·(π, π) = 3 π·(π, β) = 2
π·(π, π) = 3 π·(π, π) = 4 π·(π, π) = 3 π·(π, π) = 2 π·(π, β) = 1
π·(π, π) = 1 π·(π, β) = 2
π·(π, β) = 1
: 4 (a-ab-b-bc-c-cd-d-da-a) : 84 πππ : π (πΊ) = π(π₯) = 1 π₯
π·(π, π) = 1 π·(π, π) = 2 π·(π, π) = 1 π·(π, β) = 2
: π·(πΊ) =
Diameter
πππ₯ π(π₯) = 5 π₯
4) Tentukan Pusat pada Graf latihan 2.2.2 PUSAT ( ) a.
a
b
c
h
d g
e
f
b.
a
b c
h
d
g e
f c.
a
b c
h
d
g f d.
e
a
b
d g
j i
e.
a
b
e
f h
e
f
d
g c
i h
h
5) Temukan Pusat pada Graf latihan 2.2.3 a.
a b
f
c
d
e
b.
a
b
h
c
g
d f
c.
e
a
b c
h g
d f
e
a
b
c
d
e
f
Graf G
Temukan π(π₯)untuk setiap titik pada πΊ. JAWAB: π(π) = π·(π, π) = π·(π, π) = 3 π(π) = π·(π, π) = π·(π, π) = 2 π(π) = π·(π, π) = π·(π, π) = 3 π(π) = π·(π, π) = 3 π(π) = π·(π, π) = 3 π(π) = π·(π, π) = π·(π, π) = 2
DIAMETER Diameter di graf πΊ adalah nilai maksimum dari eksentisitas titik-titik di πΊ. Simbol : π·(πΊ) =
ππππ π(π₯) π₯
Dari contoh 1 , diperoleh diameter π·(πΊ) = 3, yang merupakan nilai terbesar dari eksentrisitas graf G.
RADIUS Radius di graf πΊ adalah nilai terkecil dari eksentisitas titik-titik di πΊ. Simbol : π (πΊ) =
πππ π(π₯) π₯
Dari contoh 1 diperoleh Radius π (πΊ) = 2, yang merupakan nilai terkecil dari eksentrisitas graf G.
LATIHAN SOAL 1) Temukan Lilitan,Radius, dan Diameter graf dibawah. (i) πΏπ jawab : π(π₯) = {4,3,5,7,6,6,5,4,4,6,5,6,6,5,5} Ukuran :1 πππ Radius : π (πΊ) = π(π₯) = 3 π₯ ππππ Diameter : π·(πΊ) = π(π₯) = 7 π₯ (ii) Graf Petersen Jawab: π(π₯) = {1,2,3,2,3,2,3,1,2,1,2,3,2,1,3,2,1,1,1,2,4,1,3,2} Ukuran :3 πππ Radius : π (πΊ) = π(π₯) = 1 π₯ πππ₯ Diameter : π·(πΊ) = π(π₯) = 4 π₯ (iii)
Prisma Segilima Jawab: π(π₯) = {5,6,8,9} Ukuran :9 πππ π(π₯) = 5 π₯ πππ₯ : π·(πΊ) = π(π₯) = 9 π₯ : π (πΊ) =
Radius Diamater
2) Tentukan Lintasan, Keliling, Radius, dan Diameter dari graf berikut. i.
a
b
c
h
d g
Jawab: π·(π, π) = 1 π·(π, π) = 2 π·(π, π) = 2 π·(π, π) = 3 π·(π, π) = 2 π·(π, π) = 2 π·(π, β) = 1 π·(π, π) = 1 π·(π, π) = 2
f
e π·(π, π) = 1 π·(π, π) = 1 π·(π, π) = 2 π·(π, π) = 2 π·(π, π) = 2 π·(π, β) = 1
π·(π, π) = 1 π·(π, π) = 2 π·(π, π) = 2 π·(π, π) = 3 π·(π, β) = 2
π·(π, π) = 1 π·(π, β) = 1
π·(π, β) = 1
π·(π, π) = 1 π·(π, π) = 1 π·(π, π) = 1 π·(π, β) = 2
π·(π, β) = 2 Lilitan Keliling
: 3 (a-ab-b-bh-h-ha-a) :8 πππ : π (πΊ) = π(π₯) = 1 π₯ πππ₯ : π·(πΊ) = π(π₯) = 3 π₯
Radius Diameter ii.
a
b c
h
d
g f Jawab: π·(π, π) = 1 π·(π, π) = 2 π·(π, π) = 3 π·(π, π) = 4 π·(π, π) = 3 π·(π, π) = 2 π·(π, β) = 1 π·(π, π) = 1 π·(π, π) = 2 π·(π, β) = 3 Lilitan Keliling Radius Diameter
e π·(π, π) = 1 π·(π, π) = 2 π·(π, π) = 3 π·(π, π) = 1 π·(π, π) = 2 π·(π, β) = 2
π·(π, π) = 1 π·(π, π) = 2 π·(π, π) = 2 π·(π, π) = 3 π·(π, β) = 3
π·(π, π) = 1 π·(π, β) = 2
π·(π, β) = 1
: 5 (a-ab-b-bf-f-fg-g-gh-h-ha-a) :8 πππ : π (πΊ) = π(π₯) = 1 π₯ πππ₯ : π·(πΊ) = π(π₯) = 4 π₯
π·(π, π) = 1 π·(π, π) = 2 π·(π, π) = 3 π·(π, β) = 4
iii.
a
b c
h
d
g f
e
Jawab: π·(π, π) = 1 π·(π, π) = 2 π·(π, π) = 3 π·(π, π) = 4 π·(π, π) = 3 π·(π, π) = 2 π·(π, β) = 1 π·(π, π) = 1 π·(π, π) = 2 π·(π, β) = 3 Lilitan Keliling
π·(π, π) = 1 π·(π, π) = 2 π·(π, π) = 3 π·(π, π) = 4 π·(π, π) = 3 π·(π, β) = 2
π·(π, π) = 1 π·(π, π) = 2 π·(π, π) = 3 π·(π, π) = 2 π·(π, β) = 1
π·(π, π) = 1 π·(π, β) = 2
π·(π, β) = 1
π·(π, π) = 1 π·(π, π) = 2 π·(π, π) = 3 π·(π, β) = 2
: 4 (a-ab-b-bc-c-ch-h-ha-a) :8 πππ : π (πΊ) = π(π₯) = 1 π₯ πππ₯ : π·(πΊ) = π(π₯) = 4 π₯
Radius Diameter iv.
j
a
b
e
f
d
g i c
Jawab: π·(π, π) = 1 π·(π, π) = 2 π·(π, π) = 3 π·(π, π) = 4
h π·(π, π) = 1 π·(π, π) = 2 π·(π, π) = 3 π·(π, π) = 4
π·(π, π) = 1 π·(π, π) = 2 π·(π, π) = 3 π·(π, π) = 2
π·(π, π) = 1 π·(π, π) = 2 π·(π, π) = 1 π·(π, β) = 2
π·(π, π) = 5 π·(π, π) = 4 π·(π, β) = 3 π·(π, π) = 4 π·(π, π) = 5
π·(π, π) = 3 π·(π, β) = 2 π·(π, π) = 3 π·(π, π) = 4
π·(π, β) = 1 π·(π, π) = 2 π·(π, π) = 3
π·(π, π) = 3 π·(π, π) = 4
π·(π, π) = 1 π·(π, π) = 2 π·(π, β) = 3 π·(π, π) = 4 π·(π, π) = 5
π·(π, π) = 1 π·(π, β) = 2 π·(π, π) = 3 π·(π, π) = 4
π·(π, β) = 1 π·(π, π) = 2 π·(π, π) = 3
π·(β, π) = 1 π·(β, π) = 2
π·(π, π) = 1 Lilitan Keliling
: 6 (e-ef-f-fg-g-gh-h-hc-c-cd-d-de-d) :9 πππ : π (πΊ) = π(π₯) = 1 π₯ πππ₯ : π·(πΊ) = π(π₯) = 5 π₯
Radius Diameter v.
a
c
b g
j i
d f
e
h
Jawab: π·(π, π) = 2 π·(π, π) = 3 π·(π, π) = 4 π·(π, π) = 3 π·(π, π) = 3 π·(π, π) = 1 π·(π, β) = 2 π·(π, π) = 2 π·(π, π) = 3
π·(π, π) = 1 π·(π, π) = 2 π·(π, π) = 2 π·(π, π) = 1 π·(π, π) = 1 π·(π, β) = 2 π·(π, π) = 2 π·(π, π) = 3
π·(π, π) = 1 π·(π, π) = 1 π·(π, π) = 2 π·(π, π) = 2 π·(π, β) = 2 π·(π, π) = 3 π·(π, π) = 4
π·(π, π) = 2 π·(π, π) = 3 π·(π, π) = 3 π·(π, β) = 3 π·(π, π) = 4 π·(π, π) = 5
π·(π, π) = 2 π·(π, π) = 2 π·(π, β) = 1 π·(π, π) = 3
π·(π, π) = 2 π·(π, β) = 1 π·(π, π) = 3 π·(π, π) = 4
π·(π, β) = 1 π·(π, π) = 1 π·(π, π) = 2
π·(β, π) = 2 π·(β, π) = 3
π·(π, π) = 4 π·(π, π) = 1 Lilitan Keliling
: 4 (g-gb-b-bf-f-fh-h-hg-g) :5 πππ : π (πΊ) = π(π₯) = 1 π₯ πππ₯ : π·(πΊ) = π(π₯) = 5 π₯
Radius Diameter
3) Tentukan Lintasan, Keliling, radius dan diameter graf berikut. a.
a b c
f d
e
Jawab: π·(π, π) = 1 π·(π, π) = 2 π·(π, π) = 2 π·(π, π) = 2 π·(π, π) = 1
Lilitan Keliling
π·(π, π) = 1 π·(π, π) = 1 π·(π, π) = 2 π·(π, π) = 2
: 3 (a-ab-b-bf-f-fa-a) :6 πππ : π (πΊ) = π(π₯) = 1 π₯ πππ₯ : π·(πΊ) = π(π₯) = 2 π₯
Radius Diameter
b.
a
b c
h
d
g f
e
π·(π, π) = 1 π·(π, π) = 2 π·(π, π) = 2
π·(π, π) = 1 π·(π, π) = 1
Jawab: π·(π, π) = 1 π·(π, π) = 2 π·(π, π) = 3 π·(π, π) = 4 π·(π, π) = 3 π·(π, π) = 2 π·(π, β) = 1 π·(π, π) = 1 π·(π, π) = 2 π·(π, β) = 3 Lilitan Keliling
π·(π, π) = 1 π·(π, π) = 2 π·(π, π) = 3 π·(π, π) = 4 π·(π, π) = 3 π·(π, β) = 2
π·(π, π) = 1 π·(π, π) = 2 π·(π, π) = 3 π·(π, π) = 2 π·(π, β) = 1
π·(π, π) = 1 π·(π, β) = 2
π·(π, β) = 1
π·(π, π) = 1 π·(π, π) = 2 π·(π, π) = 1 π·(π, β) = 2
: 4 (a-ab-b-bc-c-ch-h-ha-a) :8 πππ : π (πΊ) = π(π₯) = 1 π₯ πππ₯ : π·(πΊ) = π(π₯) = 4 π₯
Radius Diameter c.
a
b c
h g
d f
Jawab: π·(π, π) = 1 π·(π, π) = 2 π·(π, π) = 3 π·(π, π) = 4 π·(π, π) = 3 π·(π, π) = 2 π·(π, β) = 1 π·(π, π) = 1 π·(π, π) = 2 π·(π, β) = 3 Lilitan Keliling Radius
e π·(π, π) = 1 π·(π, π) = 4 π·(π, π) = 5 π·(π, π) = 4 π·(π, π) = 3 π·(π, β) = 2
π·(π, π) = 3 π·(π, π) = 4 π·(π, π) = 3 π·(π, π) = 2 π·(π, β) = 1
π·(π, π) = 1 π·(π, β) = 2
π·(π, β) = 1
: 4 (a-ab-b-bc-c-cd-d-da-a) : 84 πππ : π (πΊ) = π(π₯) = 1 π₯
π·(π, π) = 1 π·(π, π) = 2 π·(π, π) = 1 π·(π, β) = 2
: π·(πΊ) =
Diameter
πππ₯ π(π₯) = 5 π₯
4) Tentukan Pusat pada Graf latihan 2.2.2 PUSAT ( ) a.
a
b
c
h
d g
e
f
b.
a
b c
h
d
g e
f c.
a
b c
h
d
g f d.
e
a
b
d g
j i
e.
a
b
e
f h
e
f
d
g c
i h
h
5) Temukan Pusat pada Graf latihan 2.2.3 a.
a b
f
c
d
e
b.
a
b
h
c
g
d f
c.
e
a
b c
h g
d f
e
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