Index of Notation

SETS AND NUMBERS SUBGRAPHS

e base of natural logarithm, 12

[x] ceiling [ij floor [n] { l , 2 , . . . , n } , l [X]k fc-element subsets of X,

5 [n]* ¿-element subsets of [n],

5 n!! semi-factorial, 140 {x)k descending factorial,

144

V E R T I C E S A N D E D G E S

V(G) VGMG) E(G) ec,e(G) eG(V)

eG(A,B)

v{R,H)

e(R,H)

vertex set, 6 number of vertices, 6 edge set, 6 number of edges, 6 number of edges within

V, 7 number of edges between

A and B, 7 number of extension

vertices, 281 number of proper edges,

281

G[V)

G[E] E(G) £(G) («,G) (R,G) clt(W) cr*(G) cr(G) ker(G)

D E N S I T I E S

induced, or spanned subgraph, 7

spanning subgraph, 7 subgraph plot, 63 roof of subgraph plot, 63 rooted graph, 68 rooted graph, 73, 281 t-closure, 282 fc-core, 106 2-core, 122 kernel, 122

d(G) m(G)

dW(G) mW(G)

dW(G) m<2>(G)

d(v,G) m(v, G)

density, 6, 64 maximum density, 6, 56,

64 K\ -density, 64 maximum K¡ -density,

64, 197 KVdensity, 65 maximum ^-density,

65 rooted density, 69 maximum rooted density,

69

327

Random Graphs by Svante Janson, Tomasz Luczak and Andrzej Rucinski

Copyright © 2000 John Wiley & Sons, Inc.

328 INDEX OF NOTATION

d(R,G) m(R,G)

P(G) PM

d„{U,W) d..H(U,W)

rooted density, 74 maximum rooted density,

74 relative density, 7 relative density of

G(n, M), 222 pair density, 213 scaled pair density, 212

DEGREES AND NEIGHBORS

NG{v) NG{S) NG{v)

~ÑG(S)

Í(C) A(G) deg(u)

neighborhood of υ, 7 neighborhood of S, 7 closed neighborhood of υ,

7 closed neighborhood of

S, 7 minimum degree, 7 maximum degree, 7 vertex degree, 7

SPECIAL GRAPHS

Gc

Pk

Kl.n JG

ΚΓ

null graph, also empty set, 7

complement of G, 79 complete graph, 7 complete bipartite graph,

7 cycle, 7 path with fc edges, 7 star, 7 union of disjoint copies, 7 matching, 7 whisk graph, 68 lollipop graph, 71 diamond, 97

GRAPH PARAMETERS

aut(G)

a(G)

X(G) D(G) ex(F.G) ex(F,G)

number of automorphisms, 7

stability, or independence number, 7

chromatic number, 7 degeneracy number, 7 Turan number, 204 relative Turan number,

204

GRAPH PROPERTIES

COVG covering property, 68 Ext(Ä, G) extension statement, 73 PM perfect matching

property, 84

Fc(e)

F -> (G)»

F -> ( O ?

M*

partial G'-fattor property, 91

vertex Ramsey property, 196

edge Ramsey property, 202

Hamilton-matching property, 105

P R O B A B I L I T Y

P 1[£] VX

VX! Xk

L

M O M E N T S

E Var Cov E(X | S)

m X

E X * E(X)t x*(X)

probability, 1 indicator function, 8 characteristic function,

145 joint characteristic

function, 147 dependency graph, 11

expectation, 8 variance, 8 covariance, 8 conditional expectation,

8 median, 40 standardized random

variable, 139 moments, 140 factorial moments, 144 cumulante, 145

κ(Χι , . . . , Xfc) mixed cumulants, 147

DISTRIBUTIONS

c d

->

Bi(n,p) Be(p) Po(A) Ν(μ,<τ2) d T V ( X , V )

d i ( X , V )

distribution, 7 convergence in

distribution, 8 convergence in

probability, 8 binomial distribution, 7 Bernoulli distribution, 7 Poisson distribution, 7 normal distribution, 7 total variation distance,

153 distance between

distributions, 158

A S Y M P T O T I C S

an = 0(6„) a„ = Ω(6„) a„ = θ(6„)

an x i»n

big O, 9 inverse big O, 9 same order of magnitude,

10 same as αη = θ(6„) , 10

INDEX OF NO TA TION 329

On = o(6„) an < bn

On > 6n a.a.s.

asymptotic equality, 10 little o, 10 same as an = o(6„), 10 same as 6„ = o(an), 10 asymptotically almost

surely, 10

S U B G R A P H C O U N T S

P R O B A B I L I T Y A S Y M P T O T I C S

Xn = Op(an) probabilistic big O, 10 Xn = Oc(an) stronger probabilistic big

O, 10 Xn = θρ(αη) probabilistic Θ(α η ) , 10 Xn = ©c(<>n) stronger probabilistic

θ ( α η ) , 10 X n = Op(a„) probabilistic little o, 11

R A N D O M S T R U C T U R E S

Γ ρ binomial random subset, 5

ΓΜ uniform random subset, 5

Fpi...,ΡΛΓ general random subset, 6

{ Γ Μ } Μ random subset process, 13

G(n, p) binomial random graph, 2

G(m, n ,p ) bipartite random graph, 2

G(n, M) uniform random graph, 3

€(*, ¿) connected random graph, 123

G(n, r) random regular graph, 3, 233

G*(n,r) random regular multigraph, 235

G ' (n , r ) random regular multigraph without

_ loops, 257 G(n,p) special random graph,

296 GL (n, M) G(n, M) without largest

component, 130 {G(t)}( random graph process, 4 {G(n, M)}M the random graph

process, 4 Gn « G„ contiguity of random

graphs, 257 G} + Gj sum of random graphs,

257 Gi φ G2 simple sum of random

graphs, 257 P(n) random permutation, 263

XG Φ<7

YG

Xc

TG

r„ Sn(H)

zk

Xn(H)

X'JH)

Hn

Un t r (n ,M)

Y(k,t)

C(kJ)

κ(η,Μ)

subgraph count, 55 minimum expected

subgraph count, 56 induced subgraph count, 7 subgraph count in

G(n, M ) , 61 isolated subgraph count,

79 isolated υ-vertex trees

count, 80 "centralized" subgraph

count, 165 cycle count in G(n, r)

a n d G * ( n , r ) , 236 decomposition

coefficients, 166 scaled decomposition

coefficients, 168 Hamilton cycle count in

G ( n , r ) , 2 4 0 Hamilton cycle count in

G*(n , r ) , 240 size of r- th largest

component, 112 ¿-component count in

G(n, M), 113 number of connected

graphs, 113 excess of largest

component of G(n, M), 121

78

T H R E S H O L D S

M M δ(ε)

LOGIC

threshold in G(n, p), 18 threshold in G{n, Λ/), 18 hitting t ime, 19 width of threshold, 20

L~

¿ord

X ~y qd(v) Thf c(M)

M \=φ

first-order language of graphs, 272

first-order language of ordered graphs, 272

adjacency predicate, 272 quantifier depth, 272 set of sentences of depth

at most k, 273 M is a model for φ, 273

E h r * ( M ' . M " ) Ehrenfeucht game, 274 M1 φ Μ2 Gi + G 2 T

sum scheme of models, 293 sum scheme of graphs, 293 signature, 293