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Table 1 Notation used in this work

From: Non-backtracking cycles: length spectrum theory and graph mining applications

Symbol

Definition

π1(X,p)

The fundamental group of X with basepoint p

[c]

Homotopy class of loop c

\(\mathcal {L}, \mathcal{L'}\)

Length spectrum function, relaxed length spectrum

Conv(X)

If X is a graph, Conv(X) is its 2-core

G=(V,E)

An undirected graph with node set V and edge set E

n,m

Number of nodes and number of edges of a graph G

e, e−1

A directed edge e=uv and its inverse e−1=vu

NBC

Non-backtracking cycle, in which no edge is followed by its inverse

B

2m×2m non-backtracking matrix of a graph

B

2n×2n matrix whose eigenvalues are the same as those of B, save for ±1

λk=ak+ibk

k-th largest eigenvalue, in magnitude, of B

Re(λ),Im(λ)

Real and imaginary parts of the complex number λ

P,Q

n×2m directed incidence matrices of a graph

p k

The fraction of a graphs’ nodes with degree k

γ

The degree exponent in the case when pkkγ

ki

i-th moment of the degree distribution pk

deg(u)

Degree of node u

nnz(A)

Number of non-zero elements of binary matrix A

NBD(G,H)

Non-backtracking distance between graphs G and H

d({λk},{μk})

Distance between eigenvalues λk and eigenvalues μk

r

Number of eigenvalues computed from one graph

\(\rho = \sqrt {\lambda _{1}}\)

Magnitude threshold for eigenvalue computation

r 0

Number of eigenvalues whose magnitude is greater than ρ

σ

Spread parameter of the RBF kernel