$\mathbf{A}$	Adjacency matrix of a graph $\mathcal{G}$.
$a_{ij}$	Element of the adjacency matrix $\mathbf{A}$
$\overline{\alpha }\left(t\right)$	Average return probability
$\langle \mathcal{C}\rangle$	Mean clustering coefficient of a network.
$\mathbf{D}$	Node degree matrix: diag $(k_1,\cdots ,k_N)$. It is the diagonal matrix formed from the nodes degrees.
$d_E(i,j)$	Euclidean distance between any pair of nodes i and j in a network.
$d_{ij}$	Distance between two nodes i and j. It is the length of the shortest path (geodesic path) between them, that is, the minimum number of links when going from one node to the other.
$d_{E,Lim}$	$d_{E,Lim}\equiv d_S$ Euclidean distance limit beyond which there is no link formation.
$E_{QD}$	Discrete electron energy in a quantum dot (QD).
${\eta }_{QT}$	Quantum transport efficiency.
$\mathcal{G}$	Graph $\mathcal{G}\equiv \mathcal{G}(\mathcal{N},\mathcal{L},{\mathbf{W}}_{PA})$, where $\mathcal{N}$ is the set of nodes (card$\left(\mathcal{N}\right)=N$), $\mathcal{L}$ is the set of links, and ${\mathbf{W}}_{PA}$ is weighted adjacency matrix that emerges from our method to link formation.
$\widehat{H}$	Hamiltonian operator corresponding to the total energy of a quantum system.
$\mathbf{H}$	Hamiltonian in matrix form.
h	Planck constant.
ℏ	Reduced Planck constant.
$|i\rangle$	Ket vector in the Hilbert space $\mathcal{H}$. It corresponds to the electron wave function in nanostructure (≡ site ≡ node ≡ ket) i.
$\langle i|$	Bra vector in the dual space corresponding to the ket $|i\rangle$ $\in \mathcal{H}$
$\langle k\rangle$	Average node degree.
$k_i$	Degree of a node i. It is the number of links connecting i to any other node.
ℓ	Average path length of a network. It is the mean value of distances between any pair of nodes in the network.
$\mathcal{L}$	Set of links (edges) of a network (graph).
$\mathbf{L}$	Laplacian matrix of a graph $\mathcal{G}$.
${\mathcal{L}}_N$	Normalized Laplacian matrix, ${\mathcal{L}}_N=$ ${\mathbf{D}}^{-1/2}\mathbf{L}{\mathbf{D}}^{-1/2}$.
m	Electron mass.
M	Size of a graph $\mathcal{G}$. It is the number of links in the set $\mathcal{L}$.
N	Order of a graph $\mathcal{G}=(\mathcal{N},\mathcal{L})$. It is the number of nodes in set $\mathcal{N}$, that is, the cardinality of set $\mathcal{N}$: $N=\left|\mathcal{N}\right|\equiv \mathrm{card}\left(\mathcal{N}\right)$.
$\mathcal{N}$	Set of nodes (or vertices) of a graph.
${▿}^2$	Laplace operator.
${\mathsf{P}}_{j\rightsquigarrow k}$	Probability for an electron to evolve between kets $|j\rangle$ and $|k\rangle$ in the time interval t.
$P\left(k\right)$	Probability density function giving the probability that a randomly selected node has k links.
$|\psi \rangle$	Ket or vector state in Dirac notation corresponding to the wave function $\psi$.
$R_{QD}$	Radius of the quantum dot.
${\psi }_{QD}$	Electron wavefunction in a quantum dot.
$S_{GC}$	$S_{GC}=N_{GC}/N$ normalized size of the giant component (GC) with respect to the total number of nodes N.
$s_{A{P}_i}$	Sum of the probability amplitudes on ket $|i\rangle$, $s_{A{P}_i}\equiv$ ${\sum }_{i\ne j}{\left({\mathbf{W}}_{PA}\right)}_i={\sum }_{i\ne j}\langle i|j\rangle$.
$\widehat{V}$	Potential energy operator.
$-V_C$	Depth of confinement potential.
$U_C\left(r\right)$	Confining, spherical (depending only on the radial co-ordinate r), finite, and “square” potential energy.
${\widehat{U}}_{{\mathcal{L}}_N}\left(t\right)$	Time evolution operator generated by the normalized Laplacian matrix ${\mathcal{L}}_N$.
$w_{ij}$	Weight of the link between node i and j. We define it as the overlap integral between the electron wave functions in kets i and j or the probability amplitude $\langle i|j\rangle$.
${\mathbf{W}}_{PA}$	weighted adjacency matrix whose elements are quantum probability amplitudes.