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Table 1 Notation Summary

From: A family of tractable graph metrics

[n]Set {1,…,n}
\(\mathbb {R}^{n\times n} \)The set of real n×n matrices.
\(\mathbb {S}^{n}\)The set of real, symmetric matrices.
IThe identity matrix of size n×n.
1The n-dimensional vector whose entries are all equal to 1.
σmax(·)Largest singular value of a matrix.
\(\mathop {\mathsf {tr}}(\cdot)\)The trace of a matrix.
conv(·)The convex hull of a set.
G(V,E)Graph with vertex set V and edge set E.
A,BMatrices [ai,j]i,j[n],[bi,j]i,j[n].
·pOperator or entry-wise p-norm.
·FFrobenius norm.
\(\mathbb {P}^{n} \)Set of permutation matrices of size n×n, c.f. (4)
\(\mathbb {W}^{n} \)Set of doubly stochastic matrices (a.k.a. the Birkhoff polytope) of size n×n, c.f. (5)
\(\mathbb {O}^{n} \)Set of orthofonal matrices (a.k.a. the Stiefel manifold) of size n×n, c.f. (6)
Ω, \(\tilde {\Omega }\)Sets over which a metric is defined.
d(x,y)A metric over space Ω.
\(\bar {d}(x,y)\)The symmetric extension of d(x,y).
(Ω,d)A metric space.
GA,GBGraphs with adjacency matrices A,B.
P,W,On×n matrices.
SA closed and bounded subset of \(\ensuremath {\mathbb {R}}^{n\times n}\).
dS(A,B)A class of distance scores defined by minimization (12) over set S.
\(d_{\mathbb {P}^{n} }\)Pseudometric dS, where S is the set of permutation matrices.
\(d_{\mathbb {W}^{n} }\)Pseudometric dS, where S is the set of doubly stochastic matrices.
\(d_{\mathbb {O}^{n} }\)Pseudometric dS, where S is the set of orthogonal matrices.
\(\Psi ^{n}_{\tilde {\Omega }}\)Set of all embeddings from \([n]\to \tilde {\Omega }\), where \((\tilde {\Omega },\tilde {d})\) is a metric space.
ψA,ψBEmbeddings in \(\Psi ^{n}_{\tilde {\Omega }}\) of nodes in graphs GA and GB, respectively.
\(D_{\psi _{A},\psi _{B}}\)n × n matrix of all pairwise distances between images of nodes in GA and GB, under embeddings ψA and ψB.