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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