The following list summarizes the notation used in this textbook.
x |
A scalar |
x |
A vector |
X |
A matrix |
XT |
Transpose of matrix X |
XH |
Hermitian transpose of matrix X |
I |
Identity matrix |
|x| |
Absolute value of x |
x∗ |
Complex conjugate of x |
||x||p |
ℓp-norm of vector x |
|
Inner product of vectors x and y |
|
Set of real numbers |
|
Set of integer numbers excluding zero |
|
Set of complex numbers |
|
Real n-length vectors |
|
Complex n-length vectors |
|
Real m × n matrices |
|
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 |
ωij |
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 |
di |
Degree of node i |
Average degree of a graph |
|
dmax |
Maximum degree of a graph |
d(i, j) |
Geodesic distance between nodes i and j |
|
Complement of graph |
|
Line graph of graph |
|
Diameter of graph |
Lnorm |
Normalized Laplacian matrix |
Lin |
In-degree Laplacian matrix |
Lout |
Out-degree Laplacian matrix |
Din |
In-degree matrix |
Dout |
Out-degree matrix |
f |
Graph signal vector |
f (i) |
Scalar graph signal value at node i |
λℓ |
Eigenvalue of the Laplacian matrix |
uℓ |
Eigenvector of the Laplacian matrix |
|
GFT vector of graph signal f |
|
GFT coefficient at frequency λℓ |
|
Shifted version of graph signal f |
∇i(f) |
Derivative of graph signal f at vertex i |
Sp(f) |
p-Dirichlet form of graph signal f |
TV(f) |
Total variation of graph signal f |
fTLf |
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 |
ψt,n |
Wavelet vector with scale t centered at node n |
Wf (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 |
gjk(i) |
Number of shortest paths from node j to k that pass through node i |
gjk |
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”) |
Ti |
Translation operator (to node i) |
Mk |
Modulation operator (to frequency λk) |
ρ(λ) |
Spectral density (distribution) function |
// |
Comment line in an algorithm |
j |
The imaginary number representation for complex numbers where |
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