scholarly journals Geary’s c and Spectral Graph Theory

Mathematics ◽  
2021 ◽  
Vol 9 (19) ◽  
pp. 2465
Author(s):  
Hiroshi Yamada
Keyword(s):  
Fourier Transform ◽  
Graph Theory ◽  
Spatial Autocorrelation ◽  
Correlation Coefficient ◽  
Graph Laplacian ◽  
Spectral Graph Theory ◽  
Algebraic Graph Theory ◽  
Spectral Graph ◽  
New Perspective ◽  
Geary’S C

Spatial autocorrelation, of which Geary’s c has traditionally been a popular measure, is fundamental to spatial science. This paper provides a new perspective on Geary’s c. We discuss this using concepts from spectral graph theory/linear algebraic graph theory. More precisely, we provide three types of representations for it: (a) graph Laplacian representation, (b) graph Fourier transform representation, and (c) Pearson’s correlation coefficient representation. Subsequently, we illustrate that the spatial autocorrelation measured by Geary’s c is positive (resp. negative) if spatially smoother (resp. less smooth) graph Laplacian eigenvectors are dominant. Finally, based on our analysis, we provide a recommendation for applied studies.

scholarly journals Using Spectral Graph Theory to Map Qubits onto Connectivity-limited Devices

2021 ◽  
Vol 2 (1) ◽  
pp. 1-30
Author(s):  
Joseph X. Lin ◽  
Eric R. Anschuetz ◽  
Aram W. Harrow
Keyword(s):  
Graph Theory ◽  
Graph Drawing ◽  
Undirected Graph ◽  
Quantum Algorithms ◽  
Quantum Circuit ◽  
Graph Laplacian ◽  
Spectral Graph Theory ◽  
One Dimensional ◽  
Spectral Graph ◽  
Logical Qubits

We propose an efficient heuristic for mapping the logical qubits of quantum algorithms to the physical qubits of connectivity-limited devices, adding a minimal number of connectivity-compliant SWAP gates. In particular, given a quantum circuit, we construct an undirected graph with edge weights a function of the two-qubit gates of the quantum circuit. Taking inspiration from spectral graph drawing, we use an eigenvector of the graph Laplacian to place logical qubits at coordinate locations. These placements are then mapped to physical qubits for a given connectivity. We primarily focus on one-dimensional connectivities and sketch how the general principles of our heuristic can be extended for use in more general connectivities.

scholarly journals Nordhaus–Gaddum-Type Relations for Arithmetic-Geometric Spectral Radius and Energy

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Yajing Wang ◽  
Yubin Gao
Keyword(s):  
Graph Theory ◽  
Spectral Radius ◽  
Adjacency Matrix ◽  
Upper Bounds ◽  
Simple Graph ◽  
Spectral Graph Theory ◽  
Extremal Graphs ◽  
Spectral Graph ◽  
Vertex Set

Spectral graph theory plays an important role in engineering. Let G be a simple graph of order n with vertex set V=v1,v2,…,vn. For vi∈V, the degree of the vertex vi, denoted by di, is the number of the vertices adjacent to vi. The arithmetic-geometric adjacency matrix AagG of G is defined as the n×n matrix whose i,j entry is equal to di+dj/2didj if the vertices vi and vj are adjacent and 0 otherwise. The arithmetic-geometric spectral radius and arithmetic-geometric energy of G are the spectral radius and energy of its arithmetic-geometric adjacency matrix, respectively. In this paper, some new upper bounds on arithmetic-geometric energy are obtained. In addition, we present the Nordhaus–Gaddum-type relations for arithmetic-geometric spectral radius and arithmetic-geometric energy and characterize corresponding extremal graphs.

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