On the number of minimal codewords in codes generated by the adjacency matrix of a graph

In this paper, the notion of energy extended to spherical fuzzy graph. The adjacency matrix of a spherical fuzzy graph is defined and we compute the energy of a spherical fuzzy graph as the sum of absolute values of eigenvalues of the adjacency matrix of the spherical fuzzy graph. Also, the lower and upper bounds for the energy of spherical fuzzy graphs are obtained.

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Mathematics of networks

10.1093/oso/9780198805090.003.0006 ◽

2018 ◽

Author(s):

Mark Newman

Keyword(s):

Random Walks ◽

Adjacency Matrix ◽

Graph Partitioning ◽

Dynamic Networks ◽

Graph Laplacian ◽

Network Visualization ◽

Final Part ◽

Bipartite Networks ◽

Component Structure ◽

Basic Properties

An introduction to the mathematical tools used in the study of networks. Topics discussed include: the adjacency matrix; weighted, directed, acyclic, and bipartite networks; multilayer and dynamic networks; trees; planar networks. Some basic properties of networks are then discussed, including degrees, density and sparsity, paths on networks, component structure, and connectivity and cut sets. The final part of the chapter focuses on the graph Laplacian and its applications to network visualization, graph partitioning, the theory of random walks, and other problems.

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A Note on the Estrada Index of the Aα-Matrix

Mathematics ◽

10.3390/math9080811 ◽

2021 ◽

Vol 9 (8) ◽

pp. 811

Author(s):

Jonnathan Rodríguez ◽

Hans Nina

Keyword(s):

Lower Bounds ◽

Adjacency Matrix ◽

Estrada Index

Let G be a graph on n vertices. The Estrada index of G is an invariant that is calculated from the eigenvalues of the adjacency matrix of a graph. V. Nikiforov studied hybrids of A(G) and D(G) and defined the Aα-matrix for every real α∈[0,1] as: Aα(G)=αD(G)+(1−α)A(G). In this paper, using a different demonstration technique, we present a way to compare the Estrada index of the Aα-matrix with the Estrada index of the adjacency matrix of the graph G. Furthermore, lower bounds for the Estrada index are established.

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Properties of adjacency matrix of the directed cyclic friendship graph

10.1063/5.0042158 ◽

2021 ◽

Author(s):

Nanda Anzana ◽

Siti Aminah ◽

Suarsih Utama

Keyword(s):

Adjacency Matrix

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Normalized Sombor Indices as Complexity Measures of Random Networks

Entropy ◽

10.3390/e23080976 ◽

2021 ◽

Vol 23 (8) ◽

pp. 976

Author(s):

R. Aguilar-Sánchez ◽

J. Méndez-Bermúdez ◽

José Rodríguez ◽

José Sigarreta

Keyword(s):

Random Matrix ◽

Adjacency Matrix ◽

Matrix Theory ◽

Computational Study ◽

Random Networks ◽

Geometric Graphs ◽

Theory Approach ◽

Complexity Measures ◽

Random Geometric Graphs ◽

Highly Correlated

We perform a detailed computational study of the recently introduced Sombor indices on random networks. Specifically, we apply Sombor indices on three models of random networks: Erdös-Rényi networks, random geometric graphs, and bipartite random networks. Within a statistical random matrix theory approach, we show that the average values of Sombor indices, normalized to the order of the network, scale with the average degree. Moreover, we discuss the application of average Sombor indices as complexity measures of random networks and, as a consequence, we show that selected normalized Sombor indices are highly correlated with the Shannon entropy of the eigenvectors of the adjacency matrix.

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Attention adjacency matrix based graph convolutional networks for skeleton-based action recognition

Neurocomputing ◽

10.1016/j.neucom.2021.02.001 ◽

2021 ◽

Vol 440 ◽

pp. 230-239

Author(s):

Jun Xie ◽

Qiguang Miao ◽

Ruyi Liu ◽

Wentian Xin ◽

Lei Tang ◽

...

Keyword(s):

Action Recognition ◽

Adjacency Matrix ◽

Convolutional Networks

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The normalized distance Laplacian

Special Matrices ◽

10.1515/spma-2020-0114 ◽

2021 ◽

Vol 9 (1) ◽

pp. 1-18

Author(s):

Carolyn Reinhart

Keyword(s):

Spectral Radius ◽

Characteristic Polynomial ◽

Adjacency Matrix ◽

Connected Graph ◽

Distance Matrix ◽

Laplacian Matrix ◽

Diagonal Matrix ◽

The Matrix

Abstract The distance matrix 𝒟(G) of a connected graph G is the matrix containing the pairwise distances between vertices. The transmission of a vertex vi in G is the sum of the distances from vi to all other vertices and T(G) is the diagonal matrix of transmissions of the vertices of the graph. The normalized distance Laplacian, 𝒟𝒧(G) = I−T(G)−1/2 𝒟(G)T(G)−1/2, is introduced. This is analogous to the normalized Laplacian matrix, 𝒧(G) = I − D(G)−1/2 A(G)D(G)−1/2, where D(G) is the diagonal matrix of degrees of the vertices of the graph and A(G) is the adjacency matrix. Bounds on the spectral radius of 𝒟 𝒧 and connections with the normalized Laplacian matrix are presented. Twin vertices are used to determine eigenvalues of the normalized distance Laplacian. The distance generalized characteristic polynomial is defined and its properties established. Finally, 𝒟𝒧-cospectrality and lack thereof are determined for all graphs on 10 and fewer vertices, providing evidence that the normalized distance Laplacian has fewer cospectral pairs than other matrices.

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Iterative Methods for the Computation of the Perron Vector of Adjacency Matrices

Mathematics ◽

10.3390/math9131522 ◽

2021 ◽

Vol 9 (13) ◽

pp. 1522

Author(s):

Anna Concas ◽

Lothar Reichel ◽

Giuseppe Rodriguez ◽

Yunzi Zhang

Keyword(s):

Iterative Methods ◽

Adjacency Matrix ◽

Lanczos Method ◽

Power Method ◽

Arnoldi Method ◽

Multiple Eigenvalues ◽

Computer Storage ◽

Adjacency Matrices ◽

Lanczos Methods ◽

Perron Vector

The power method is commonly applied to compute the Perron vector of large adjacency matrices. Blondel et al. [SIAM Rev. 46, 2004] investigated its performance when the adjacency matrix has multiple eigenvalues of the same magnitude. It is well known that the Lanczos method typically requires fewer iterations than the power method to determine eigenvectors with the desired accuracy. However, the Lanczos method demands more computer storage, which may make it impractical to apply to very large problems. The present paper adapts the analysis by Blondel et al. to the Lanczos and restarted Lanczos methods. The restarted methods are found to yield fast convergence and to require less computer storage than the Lanczos method. Computed examples illustrate the theory presented. Applications of the Arnoldi method are also discussed.

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Enhancing the Accuracy of Peak Hourly Demand in Bike-Sharing Systems using a Graph Convolutional Network with Public Transit Usage Data

Transportation Research Record Journal of the Transportation Research Board ◽

10.1177/03611981211012003 ◽

2021 ◽

pp. 036119812110120

Author(s):

Jung-Hoon Cho ◽

Seung Woo Ham ◽

Dong-Kyu Kim

Keyword(s):

Adjacency Matrix ◽

Public Transit ◽

Demand Forecasting ◽

The Other ◽

Convolutional Network ◽

Demand Pattern ◽

Temporal Features ◽

Unmet Demand ◽

Bike Sharing ◽

Main Component

With the growth of the bike-sharing system, the problem of demand forecasting has become important to the bike-sharing system. This study aims to develop a novel prediction model that enhances the accuracy of the peak hourly demand. A spatiotemporal graph convolutional network (STGCN) is constructed to consider both the spatial and temporal features. One of the model’s essential steps is determining the main component of the adjacency matrix and the node feature matrix. To achieve this, 131 days of data from the bike-sharing system in Seoul are used and experiments conducted on the models with various adjacency matrices and node feature matrices, including public transit usage. The results indicate that the STGCN models reflecting the previous demand pattern to the adjacency matrix show outstanding performance in predicting demand compared with the other models. The results also show that the model that includes bus boarding and alighting records is more accurate than the model that contains subway records, inferring that buses have a greater connection to bike-sharing than the subway. The proposed STGCN with public transit data contributes to the alleviation of unmet demand by enhancing the accuracy in predicting peak demand.

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