x

A scalar

x

A vector

X

A matrix

X^T

Transpose of matrix X

X^H

Hermitian transpose of matrix X

I

Identity matrix

|x|

Absolute value of x

x^∗

Complex conjugate of x

||x||_p

ℓ_p-norm of vector x

⟨?,?⟩

Inner product of vectors x and y

ℝ

Set of real numbers

ℤ+

Set of integer numbers excluding zero

ℂ

Set of complex numbers

ℝn

Real n-length vectors

ℂn

Complex n-length vectors

ℝm×n

Real m × n matrices

ℂm×n

Complex m × n matrices

?

An arbitrary graph

?

Set of vertices in a graph

ε

Set of edges in a graph

A

Adjacency matrix

C

Incidence matrix

D

Degree matrix

L

Laplacian matrix

W

Weight matrix

ω_ij

Weight of the edge from node j to i

E

Total number of edges in a graph

|ε|

Total number of edges in a graph

|?|

Total number of vertices in a graph

N

Total number of nodes in a network

d_i

Degree of node i

d¯

Average degree of a graph

d_max

Maximum degree of a graph

d(i, j)

Geodesic distance between nodes i and j

?¯

Complement of graph ?

L(?)

Line graph of graph ?

D(?)

Diameter of graph ?

L^norm

Normalized Laplacian matrix

L_in

In-degree Laplacian matrix

L_out

Out-degree Laplacian matrix

D_in

In-degree matrix

D_out

Out-degree matrix

f

Graph signal vector

f (i)

Scalar graph signal value at node i

λ_ℓ

Eigenvalue of the Laplacian matrix

u_ℓ

Eigenvector of the Laplacian matrix

?^

GFT vector of graph signal f

f^(λℓ)

GFT coefficient at frequency λ_ℓ

?̃

Shifted version of graph signal f

∇_i(f)

Derivative of graph signal f at vertex i

S_p(f)

p-Dirichlet form of graph signal f

TV(f)

Total variation of graph signal f

f^TLf

Laplacian quadratic form of graph signal f

J

Jordan matrix

V

Matrix whose columns are eigenvectors of a graph structure matrix

U

Matrix whose columns are the eigenvectors of the Laplacian matrix

S

Shift operator

H

Graph filter matrix

h(L)

Polynomial of L

ψ_t,n

Wavelet vector with scale t centered at node n

W_f (t, n)

SGWT of graph signal f at scale t and at node n

DC(i)

Degree centrality of node i

CC(i)

Closeness centrality of node i

g_jk(i)

Number of shortest paths from node j to k that pass through node i

g_jk

Total number of shortest paths from node j to k

BC(i)

Betweenness centrality of node i

EC(i)

Eigenvector centrality of node i

⊗

Kronecker product

⊙

Element-wise Hadamard operator

×

Cartisian product

⊠

Strong product

?(⋅)

Big O growth function (O pronounced as “Oh”)

T_i

Translation operator (to node i)

M_k

Modulation operator (to frequency λ_k)

ρ(λ)

Spectral density (distribution) function

//

Comment line in an algorithm

j

The imaginary number representation for complex numbers where j=−1