$\mathcal{S}$

Set

$\overline{\mathcal{S}}$

Set complement

$\left|\mathcal{S}\right|$

Set cardinality

A

Matrix

v

Vector

$\tilde{p}$

Tuple

$\mathcal{V}$

Set of nodes

$ℰ$

Set of edges

$ℱ$

Set of cycles (faces)

$\mathcal{G}=\left(\mathcal{V},ℰ\right)$

Graph

${\upsilon }_i\in \mathcal{V}$

Nodes

υ_i ~ υ_j

Adjacency of two nodes

e_i

An edge in the context of an element of $ℰ$ (i.e., $e_i\in ℰ$)

e_ij

An undirected edge between nodes υ_i to υ_j

e_i→j

Directed edges

w(υ_i) or w_i

Node weights

w(υ_i, υ_j) or w_ij

Edge weights

x_i for node υ_i

Node variable

y_i for edge e_i

Edge variable (flow)

deg (υ_i) or d_i

Node degree

deg⁺ (υ_i)

Node in-degree

deg⁻ (υ_i)

Node out-degree

K_n

Complete graph of n vertices

K_{m, n}

Complete bipartite graph

A

Incidence matrix

W

Adjacency matrix

D

Degree matrix

L = D − W

Laplacian matrix

$\tilde{L}\text{=}{\mathcal{D}}^{-\frac{\text{1}}{\text{2}}}L{\mathcal{D}}^{-\frac{\text{1}}{\text{2}}}$

Normalized Laplacian matrix

π = (υ₁, ⋯, υ_n)

Path

ψ

Coupling