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Journal articles on the topic 'Normalized laplacian matrix'

Author: Grafiati

Published: 4 June 2025

Last updated: 15 July 2025

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1

Reinhart, Carolyn. "The normalized distance Laplacian." Special Matrices 9, no. 1 (2021): 1–18. http://dx.doi.org/10.1515/spma-2020-0114.

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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 r

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Yu, Guihai, and Hui Qu. "More on Spectral Analysis of Signed Networks." Complexity 2018 (October 16, 2018): 1–6. http://dx.doi.org/10.1155/2018/3467158.

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Spectral graph theory plays a key role in analyzing the structure of social (signed) networks. In this paper we continue to study some properties of (normalized) Laplacian matrix of signed networks. Sufficient and necessary conditions for the singularity of Laplacian matrix are given. We determine the correspondence between the balance of signed network and the singularity of its Laplacian matrix. An expression of the determinant of Laplacian matrix is present. The symmetry about 1 of eigenvalues of normalized Laplacian matrix is discussed. We determine that the integer 2 is an eigenvalue of n

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El Seidy, Essam, Salah Eldin Hussein, and Atef Mohamed. "Properties of the characteristic polynomials and spectrum of Pn and Cn." International Journal of Applied Mathematical Research 5, no. 2 (2016): 132. http://dx.doi.org/10.14419/ijamr.v5i2.6106.

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We consider a finite undirected and connected simple graph with vertex set and edge set .We calculated the general formulas of the spectra of a cycle graph and path graph. In this discussion we are interested in the adjacency matrix, Laplacian matrix, signless Laplacian matrix, normalized Laplacian matrix, and seidel adjacency matrix.

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Yu, Guihai, Matthias Dehmer, Frank Emmert-Streib, and Herbert Jodlbauer. "Hermitian normalized Laplacian matrix for directed networks." Information Sciences 495 (August 2019): 175–84. http://dx.doi.org/10.1016/j.ins.2019.04.049.

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5

Lorenzen, Kate. "Cospectral constructions for several graph matrices using cousin vertices." Special Matrices 10, no. 1 (2021): 9–22. http://dx.doi.org/10.1515/spma-2020-0143.

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Abstract Graphs can be associated with a matrix according to some rule and we can find the spectrum of a graph with respect to that matrix. Two graphs are cospectral if they have the same spectrum. Constructions of cospectral graphs help us establish patterns about structural information not preserved by the spectrum. We generalize a construction for cospectral graphs previously given for the distance Laplacian matrix to a larger family of graphs. In addition, we show that with appropriate assumptions this generalized construction extends to the adjacency matrix, combinatorial Laplacian matrix

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Usha, Subramaniam, Murugesan Karthik, Marappan Jothibasu, and Vijayaragavan Gowtham. "An integrated exploration of heat kernel invariant feature and manifolding technique for 3D object recognition system." Acta Scientiarum. Technology 46, no. 1 (2023): e62608. http://dx.doi.org/10.4025/actascitechnol.v46i1.62608.

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Spectral Graph theory has been utilized frequently in the field of Computer Vision and Pattern Recognition to address challenges in the field of Image Segmentation and Image Classification. In the proposed method, for classification techniques, the associated graph's Eigen values and Eigen vectors of the adjacency matrix or Laplacian matrix created from the images are employed. The Laplacian spectrum and a graph's heat kernel are inextricably linked. Exponentiating the Laplacian eigensystem over time yields the heat kernel, which is the solution to the heat equations. In the proposed technique

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Gupta, C. K., B. Shwetha Shetty, and V. Lokesha. "On the graph of nilpotent matrix group of length one." Discrete Mathematics, Algorithms and Applications 08, no. 01 (2016): 1650009. http://dx.doi.org/10.1142/s1793830916500099.

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In this paper we construct a Cayley graph for multiplicative group of upper unitriangular [Formula: see text] matrices over [Formula: see text] mod [Formula: see text]. Also we find some topological indices, diameter, girth, spectra and energy of adjacency, Laplacian, normalized Laplacian and signless Laplacian matrix of the same graph.

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Carmona, Ángeles, Margarida Mitjana, and Enric Monsó. "Group inverse matrix of the normalized Laplacian on subdivision networks." Applicable Analysis and Discrete Mathematics, no. 00 (2020): 23. http://dx.doi.org/10.2298/aadm180420023c.

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In this paper we consider a subdivision of a given network and we show how the group inverse matrix of the normalized laplacian of the subdivision network is related to the group inverse matrix of the normalized laplacian of the initial given network. Our approach establishes a relationship between solutions of related Poisson problems on both structures and takes advantage on the properties of the group inverse matrix. As a consequence we get formulae for effective resistances and the Kirchhoff Index of the subdivision network expressed in terms of its corresponding in the base network. Final

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Kirkland, Steve, and Debdas Paul. "Bipartite subgraphs and the signless Laplacian matrix." Applicable Analysis and Discrete Mathematics 5, no. 1 (2011): 1–13. http://dx.doi.org/10.2298/aadm110205006k.

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For a connected graph G, we derive tight inequalities relating the smallest signless Laplacian eigenvalue to the largest normalized Laplacian eigenvalue. We investigate how vectors yielding small values of the Rayleigh quotient for the signless Laplacian matrix can be used to identify bipartite subgraphs. Our results are applied to some graphs with degree sequences approximately following a power law distribution with exponent value 2:1 (scale-free networks), and to a scale-free network arising from protein-protein interaction.

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Braga, R. O., V. M. Rodrigues, and R. O. Silva. "Locating Eigenvalues of a Symmetric Matrix whose Graph is Unicyclic." Trends in Computational and Applied Mathematics 22, no. 4 (2021): 659–74. http://dx.doi.org/10.5540/tcam.2021.022.04.00659.

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We present a linear-time algorithm that computes in a given real interval the number of eigenvalues of any symmetric matrix whose underlying graph is unicyclic. The algorithm can be applied to vertex- and/or edge-weighted or unweighted unicyclic graphs. We apply the algorithm to obtain some general results on the spectrum of a generalized sun graph for certain matrix representations which include the Laplacian, normalized Laplacian and signless Laplacian matrices.

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11

JIANG, TIEFENG. "EMPIRICAL DISTRIBUTIONS OF LAPLACIAN MATRICES OF LARGE DILUTE RANDOM GRAPHS." Random Matrices: Theory and Applications 01, no. 03 (2012): 1250004. http://dx.doi.org/10.1142/s2010326312500049.

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We study the spectral properties of the Laplacian matrices and the normalized Laplacian matrices of the Erdös–Rényi random graph G(n, pn) for large n. Although the graph is simple, we discover some interesting behaviors of the two Laplacian matrices. In fact, under the dilute case, that is, pn ∈ (0, 1) and npn → ∞, we prove that the empirical distribution of the eigenvalues of the Laplacian matrix converges to a deterministic distribution, which is the free convolution of the semi-circle law and N(0, 1). However, for its normalized version, we prove that the empirical distribution converges to

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12

Liu, Qun. "Spectral analysis for weighted iterated pentagonal graphs and its applications." Modern Physics Letters B 34, no. 28 (2020): 2050308. http://dx.doi.org/10.1142/s021798492050308x.

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Deterministic weighted networks have been widely used to model real-world complex systems. In this paper, we study the weighted iterated pentagonal networks. From the construction of the network, we derive recursive relations of all eigenvalues and their multiplicities of its normalized Laplacian matrix from the two successive generations of the weighted iterated pentagonal networks. As applications of spectra of the normalized Laplacian matrix, we study the Kemeny’s constant, the multiplicative degree-Kirchhoff index, and the number of weighted spanning trees and derive their exact closed-for

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Dai, Meifeng, Yufei Chen, Xiaoqian Wang, and Weiyi Su. "Spectral analysis for weighted iterated quadrilateral graphs." International Journal of Modern Physics C 29, no. 11 (2018): 1850113. http://dx.doi.org/10.1142/s0129183118501139.

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Much information about the structural properties and dynamical aspects of a network is measured by the eigenvalues of its normalized Laplacian matrix. In this paper, we aim to present a first study on the spectra of the normalized Laplacian matrix of weighted iterated quadrilateral graphs. We analytically obtain all the eigenvalues, as well as their multiplicities from two successive generations. As an example of application of these results, we then derive closed-form expressions for the multiplicative degree Kirchhoff index and the Kemeny’s constant, as well as the number of weighted spannin

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M, M. Jariya. "Results on Characteristic Vectors." International Journal of Trend in Scientific Research and Development 3, no. 5 (2019): 1733–37. https://doi.org/10.5281/zenodo.3591388.

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In this paper we have established the bounds of the extreme characteristic roots of nlap G and sLap G by their traces. Also found the bounds for n th characteristic roots of nLap G and sLap G . M M Jariya "Results on Characteristic Vectors" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-5 , August 2019, URL: https://www.ijtsrd.com/papers/ijtsrd28006.pdf

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RATHER, BILAL A., HILAL A. GANIE, and MUSTAPHA AOUCHICHE. "On normalized distance Laplacian eigenvalues of graphs and applications to graphs defined on groups and rings." Carpathian Journal of Mathematics 39, no. 1 (2022): 213–30. http://dx.doi.org/10.37193/cjm.2023.01.14.

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The normalized distance Laplacian matrix of a connected graph $ G $, denoted by $ D^{\mathcal{L}}(G) $, is defined by $ D^{\mathcal{L}}(G)=Tr(G)^{-1/2}D^L(G)Tr(G)^{-1/2}, $ where $ D(G) $ is the distance matrix, the $D^{L}(G)$ is the distance Laplacian matrix and $ Tr(G)$ is the diagonal matrix of vertex transmissions of $ G. $ The set of all eigenvalues of $ D^{\mathcal{L}}(G) $ including their multiplicities is the normalized distance Laplacian spectrum or $ D^{\mathcal{L}} $-spectrum of $G$. In this paper, we find the $ D^{\mathcal{L}} $-spectrum of the joined union of regular graphs in ter

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16

Heydari, Abbas. "The normalized Laplacian polynomial of rooted product of graphs." Discrete Mathematics, Algorithms and Applications 11, no. 04 (2019): 1950046. http://dx.doi.org/10.1142/s1793830919500460.

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Let [Formula: see text] be a simple graph with [Formula: see text] vertices and [Formula: see text] be a sequence of [Formula: see text] rooted graphs [Formula: see text]. The rooted product [Formula: see text], of [Formula: see text] by [Formula: see text] is constructed by identifying the root vertex of [Formula: see text] with the [Formula: see text]th vertex of [Formula: see text]. In this paper, the characteristic polynomial of the normalized Laplacian matrix of [Formula: see text] is obtained. As an application of our results, we obtain the normalized Laplacian polynomial and spectrum of

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17

CHEN, YUFEI, MEIFENG DAI, XIAOQIAN WANG, YU SUN, and WEIYI SU. "SPECTRAL ANALYSIS FOR WEIGHTED ITERATED TRIANGULATIONS OF GRAPHS." Fractals 26, no. 01 (2018): 1850017. http://dx.doi.org/10.1142/s0218348x18500172.

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Much information about the structural properties and dynamical aspects of a network is measured by the eigenvalues of its normalized Laplacian matrix. In this paper, we aim to present a first study on the spectra of the normalized Laplacian of weighted iterated triangulations of graphs. We analytically obtain all the eigenvalues, as well as their multiplicities from two successive generations. As an example of application of these results, we then derive closed-form expressions for their multiplicative Kirchhoff index, Kemeny’s constant and number of weighted spanning trees.

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Chen, Yufei, and Wenxia Li. "Spectral analysis for weighted iterated q-triangulations of graphs." International Journal of Modern Physics C 31, no. 03 (2020): 2050042. http://dx.doi.org/10.1142/s0129183120500424.

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Much information about the structural properties and dynamical aspects of a network is measured by the eigenvalues of its normalized Laplacian matrix. In this paper, we aim to present a first study on the spectra of the normalized Laplacian of weighed iterated [Formula: see text]-triangulations of graphs. We analytically obtain all the eigenvalues, as well as their multiplicities from two successive generations. As examples of application of these results, we then derive closed-form expressions for their Kemeny’s constant and multiplicative Kirchhoff index. Simulation example is also provided

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19

Braga, Rodrigo Orsini, and Virgínia Maria Rodrigues. "Locating Eigenvalues of Perturbed Laplacian Matrices of Trees." TEMA (São Carlos) 18, no. 3 (2018): 479. http://dx.doi.org/10.5540/tema.2017.018.03.479.

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We give a linear time algorithm to compute the number of eigenvalues of any perturbedLaplacian matrix of a tree in a given real interval. The algorithm can be applied to weightedor unweighted trees. Using our method we characterize the trees that have up to $5$ distincteigenvalues with respect to a family of perturbed Laplacian matrices that includes the adjacencyand normalized Laplacian matrices as special cases, among others.

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20

Ju, Tingting, Meifeng Dai, Changxi Dai, Yu Sun, Xiangmei Song, and Weiyi Su. "Applications of Laplacian spectrum for the vertex–vertex graph." Modern Physics Letters B 33, no. 17 (2019): 1950184. http://dx.doi.org/10.1142/s0217984919501847.

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Complex networks have attracted a great deal of attention from scientific communities, and have been proven as a useful tool to characterize the topologies and dynamics of real and human-made complex systems. Laplacian spectrum of the considered networks plays an essential role in their network properties, which have a wide range of applications in chemistry and others. Firstly, we define one vertex–vertex graph. Then, we deduce the recursive relationship of its eigenvalues at two successive generations of the normalized Laplacian matrix, and we obtain the Laplacian spectrum for vertex–vertex

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Dai, Meifeng, Tingting Ju, Jingyi Liu, Yu Sun, Xiangmei Song, and Weiyi Su. "Applications of Laplacian spectrum for the weighted scale-free network with a weight factor." International Journal of Modern Physics B 32, no. 32 (2018): 1850353. http://dx.doi.org/10.1142/s0217979218503538.

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Laplacian spectrum gives a lot of useful information about complex structural properties and relevant dynamical aspects, which has attracted the attention of mathematicians. We introduced the weighted scale-free network inspired by the binary scale-free network. First, the weighted scale-free network with a weight factor is constructed by an iterative way. In the next step, we use the definition of eigenvalue and eigenvector to obtain the recursive relationship of its eigenvalues and multiplicities at two successive generations. Through analysis of eigenvalues of transition weight matrix we fi

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22

Liu, Jia-Bao, Qian Zheng, and Sakander Hayat. "The Normalized Laplacians, Degree-Kirchhoff Index, and the Complexity of Möbius Graph of Linear Octagonal-Quadrilateral Networks." Journal of Mathematics 2021 (October 12, 2021): 1–25. http://dx.doi.org/10.1155/2021/2328940.

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The normalized Laplacian plays an indispensable role in exploring the structural properties of irregular graphs. Let L n 8,4 represent a linear octagonal-quadrilateral network. Then, by identifying the opposite lateral edges of L n 8,4 , we get the corresponding Möbius graph M Q n 8,4 . In this paper, starting from the decomposition theorem of polynomials, we infer that the normalized Laplacian spectrum of M Q n 8,4 can be determined by the eigenvalues of two symmetric quasi-triangular matrices ℒ A and ℒ S of order 4 n . Next, owing to the relationship between the two matrix roots and the coef

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23

Boukrab, Rachid, and Alba Pagès-Zamora. "Random-Walk Laplacian for Frequency Analysis in Periodic Graphs." Sensors 21, no. 4 (2021): 1275. http://dx.doi.org/10.3390/s21041275.

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This paper presents the benefits of using the random-walk normalized Laplacian matrix as a graph-shift operator and defines the frequencies of a graph by the eigenvalues of this matrix. A criterion to order these frequencies is proposed based on the Euclidean distance between a graph signal and its shifted version with the transition matrix as shift operator. Further, the frequencies of a periodic graph built through the repeated concatenation of a basic graph are studied. We show that when a graph is replicated, the graph frequency domain is interpolated by an upsampling factor equal to the n

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24

Yin, Lifeng, Lei Lv, Dingyi Wang, Yingwei Qu, Huayue Chen, and Wu Deng. "Spectral Clustering Approach with K-Nearest Neighbor and Weighted Mahalanobis Distance for Data Mining." Electronics 12, no. 15 (2023): 3284. http://dx.doi.org/10.3390/electronics12153284.

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This paper proposes a spectral clustering method using k-means and weighted Mahalanobis distance (Referred to as MDLSC) to enhance the degree of correlation between data points and improve the clustering accuracy of Laplacian matrix eigenvectors. First, we used the correlation coefficient as the weight of the Mahalanobis distance to calculate the weighted Mahalanobis distance between any two data points and constructed the weighted Mahalanobis distance matrix of the data set; then, based on the weighted Mahalanobis distance matrix, we used the K-nearest neighborhood (KNN) algorithm construct s

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Prashanth, B., K. Nagendra Naik, and R. Salestina M. "Correction^*." Journal of Applied Mathematics, Statistics and Informatics 18, no. 1 (2022): 109–24. http://dx.doi.org/10.2478/jamsi-2022-0008.

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Abstract With this article in mind, we have found some results using eigenvalues of graph with sign. It is intriguing to note that these results help us to find the determinant of Normalized Laplacian matrix of signed graph and their coefficients of characteristic polynomial using the number of vertices. Also we found bounds for the lowest value of eigenvalue.

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Prashanth, B., K. Nagendra Naik, and R. Salestina M. "Restructured class of estimators for population mean using an auxiliary variable under simple random sampling scheme." Journal of Applied Mathematics, Statistics and Informatics 17, no. 2 (2021): 75–90. http://dx.doi.org/10.2478/jamsi-2021-0010.

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Abstract With this article in mind, we have found some results using eigenvalues of graph with sign. It is intriguing to note that these results help us to find the determinant of Normalized Laplacian matrix of signed graph and their coe cients of characteristic polynomial using the number of vertices. Also we found bounds for the lowest value of eigenvalue.

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Starosta, Bartłomiej, Mieczysław A. Kłopotek, Sławomir T. Wierzchoń, Dariusz Czerski, Marcin Sydow, and Piotr Borkowski. "Explainable Graph Spectral Clustering of text documents." PLOS ONE 20, no. 2 (2025): e0313238. https://doi.org/10.1371/journal.pone.0313238.

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Spectral clustering methods are known for their ability to represent clusters of diverse shapes, densities etc. However, the results of such algorithms, when applied e.g. to text documents, are hard to explain to the user, especially due to embedding in the spectral space which has no obvious relation to document contents. Therefore, there is an urgent need to elaborate methods for explaining the outcome of the clustering. We have constructed in this paper a theoretical bridge linking the clusters resulting from Graph Spectral Clustering and the actual document content, given that similarities

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Lu, Kangkang, Yanhua Yu, Hao Fei, et al. "Improving Expressive Power of Spectral Graph Neural Networks with Eigenvalue Correction." Proceedings of the AAAI Conference on Artificial Intelligence 38, no. 13 (2024): 14158–66. http://dx.doi.org/10.1609/aaai.v38i13.29326.

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In recent years, spectral graph neural networks, characterized by polynomial filters, have garnered increasing attention and have achieved remarkable performance in tasks such as node classification. These models typically assume that eigenvalues for the normalized Laplacian matrix are distinct from each other, thus expecting a polynomial filter to have a high fitting ability. However, this paper empirically observes that normalized Laplacian matrices frequently possess repeated eigenvalues. Moreover, we theoretically establish that the number of distinguishable eigenvalues plays a pivotal rol

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Afkhami, Mojgan, Mehdi Hassankhani, and Kazem Khashyarmanesh. "Distance between the spectra of graphs with respect to normalized Laplacian spectra." Georgian Mathematical Journal 26, no. 2 (2019): 227–34. http://dx.doi.org/10.1515/gmj-2017-0051.

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Abstract Let {G_{n}} and {G_{n}^{\prime}} be two nonisomorphic graphs on n vertices with spectra (with respect to the adjacency matrix) \lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{n}\quad\text{and}\quad\lambda% ^{\prime}_{1}\geq\lambda^{\prime}_{2}\geq\cdots\geq\lambda^{\prime}_{n}, respectively. Define the distance between the spectra of {G_{n}} and {G_{n}^{\prime}} as \lambda(G_{n},G^{\prime}_{n})=\sum_{i=1}^{n}(\lambda_{i}-\lambda^{\prime}_{i})% ^{2}\quad\biggl{(}\text{or use }\sum_{i=1}^{n}\lvert\lambda_{i}-\lambda^{% \prime}_{i}\rvert\biggr{)}. Define the cospectrality of {G_{n}} by

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Gayathri, M., та R. Rajkumar. "Spectra of (M,ℳ)-corona-join of graphs". Proyecciones (Antofagasta) 42, № 1 (2023): 105–24. http://dx.doi.org/10.22199/issn.0717-6279-5454.

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In this paper, we introduce the (M, ℳ)-corona-join of G and ℋk constrained by vertex subsets 𝒯, which is the union of two graphs: one is the M-generalized corona of a graph G and a family of graphs ℋk constrained by vertex subset 𝒯 of the graphs in ℋk, where M is a suitable matrix; and the other one is the ℳ -join of ℋk, where ℳ is a collection of matrices. We determine the spectra of the adjacency, the Laplacian, the signless Laplacian and the normalized Laplacian matrices of some special cases of the (M, ℳ)-corona-join of G and ℋk constrained by vertex subsets 𝒯. These results enable us to d

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Aliakbarisani, Roya, Abdorasoul Ghasemi, and M. Angeles Serrano. "Perturbation of the Normalized Laplacian Matrix for the Prediction of Missing Links in Real Networks." IEEE Transactions on Network Science and Engineering 9, no. 2 (2022): 863–74. http://dx.doi.org/10.1109/tnse.2021.3137862.

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Chen, Yinan, Wenbin Ye, and Dong Li. "Spectral Clustering Community Detection Algorithm Based on Point-Wise Mutual Information Graph Kernel." Entropy 25, no. 12 (2023): 1617. http://dx.doi.org/10.3390/e25121617.

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To address the problem that traditional spectral clustering algorithms cannot obtain the complete structural information of networks, this paper proposes a spectral clustering community detection algorithm, PMIK-SC, based on the point-wise mutual information (PMI) graph kernel. The kernel is constructed according to the point-wise mutual information between nodes, which is then used as a proximity matrix to reconstruct the network and obtain the symmetric normalized Laplacian matrix. Finally, the network is partitioned by the eigendecomposition and eigenvector clustering of the Laplacian matri

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Wu, Haoyuan, Xinyun Zhang, Peng Xu, Peiyu Liao, Xufeng Yao, and Bei Yu. "p-Laplacian Adaptation for Generative Pre-trained Vision-Language Models." Proceedings of the AAAI Conference on Artificial Intelligence 38, no. 6 (2024): 6003–11. http://dx.doi.org/10.1609/aaai.v38i6.28415.

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Vision-Language models (VLMs) pre-trained on large corpora have demonstrated notable success across a range of downstream tasks. In light of the rapidly increasing size of pre-trained VLMs, parameter-efficient transfer learning (PETL) has garnered attention as a viable alternative to full fine-tuning. One such approach is the adapter, which introduces a few trainable parameters into the pre-trained models while preserving the original parameters during adaptation. In this paper, we present a novel modeling framework that recasts adapter tuning after attention as a graph message passing process

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Kaliuzhnyi-Verbovetskyi, D., and V. Pivovarchik. "RECOVERING THE SHAPE OF A QUANTUM CATERPILLAR TREE BY TWO SPECTRA." Mechanics And Mathematical Methods 5, no. 1 (2023): 14–24. http://dx.doi.org/10.31650/2618-0650-2023-5-1-14-24.

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existence of co-spectral (iso-spectral) graphs is a well-known problem of the classical graph theory. However, co-spectral graphs exist in the theory of quantum graphs also. In other words, the spectrum of the Sturm-Liouville problem on a metric graph does not determine alone the shape of the graph. Сo-spectral trees also exist if the number of vertices exceeds eight. We consider two Sturm-Liouville spectral problems on an equilateral metric caterpillar tree with real L2 (0,l) potentials on the edges. In the first (Neumann) problem we impose standard conditions at all vertices: Neumann boundar

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Ahmad, Hilal, Abdollah Alhevaz, Maryam Baghipur, and Gui-Xian Tian. "Bounds for Generalized Distance Spectral Radius and the Entries of the Principal Eigenvector." Tamkang Journal of Mathematics 52, no. 1 (2021): 69–89. http://dx.doi.org/10.5556/j.tkjm.52.2021.3280.

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For a simple connected graph $G$, the convex linear combinations $D_{\alpha}(G)$ of \ $Tr(G)$ and $D(G)$ is defined as $D_{\alpha}(G)=\alpha Tr(G)+(1-\alpha)D(G)$, $0\leq \alpha\leq 1$. As $D_{0}(G)=D(G)$, $2D_{\frac{1}{2}}(G)=D^{Q}(G)$, $D_{1}(G)=Tr(G)$ and $D_{\alpha}(G)-D_{\beta}(G)=(\alpha-\beta)D^{L}(G)$, this matrix reduces to merging the distance spectral and distance signless Laplacian spectral theories. In this paper, we study the spectral properties of the generalized distance matrix $D_{\alpha}(G)$. We obtain some lower and upper bounds for the generalized distance spectral radius,

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Salim, Ibrahim, and A. Hamza. "Fast Feature-Preserving Approach to Carpal Bone Surface Denoising." Sensors 18, no. 7 (2018): 2379. http://dx.doi.org/10.3390/s18072379.

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We present a geometric framework for surface denoising using graph signal processing, which is an emerging field that aims to develop new tools for processing and analyzing graph-structured data. The proposed approach is formulated as a constrained optimization problem whose objective function consists of a fidelity term specified by a noise model and a regularization term associated with prior data. Both terms are weighted by a normalized mesh Laplacian, which is defined in terms of a data-adaptive kernel similarity matrix in conjunction with matrix balancing. Minimizing the objective functio

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Xia, Haisong, and Zhongzhi Zhang. "Efficient Approximation of Kemeny's Constant for Large Graphs." Proceedings of the ACM on Management of Data 2, no. 3 (2024): 1–26. http://dx.doi.org/10.1145/3654937.

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For an undirected graph, its Kemeny's constant is defined as the mean hitting time of random walks from one vertex to another chosen randomly according to the stationary distribution. Kemeny's constant exhibits numerous explanations from different perspectives and has found various applications in the field of complex networks. Due to the requirement of computing the inverse of the normalized Laplacian matrix, it is infeasible to get the accurate Kemeny's constant of large networks with millions of vertices. Existing methods either consume excessive memory space that are impractical for large-

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WU, BO, ZHIZHUO ZHANG, YINGYING CHEN, et al. "EIGENTIME IDENTITY FOR WEIGHT-DEPENDENT WALK ON A CLASS OF WEIGHTED FRACTAL SCALE-FREE HIERARCHICAL-LATTICE NETWORKS." Fractals 27, no. 08 (2019): 1950138. http://dx.doi.org/10.1142/s0218348x1950138x.

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In this paper, we construct a class of weighted fractal scale-free hierarchical-lattice networks. Each edge in the network generates [Formula: see text] connected branches in each iteration process and assigns the corresponding weight. To reflect the global characteristics of such networks, we study the eigentime identity determined by the reciprocal sum of non-zero eigenvalues of normalized Laplacian matrix. By the recursive relationship of eigenvalues at two successive generations, we find the eigenvalues and their corresponding multiplicities for two cases when [Formula: see text] is even o

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Gong, Yi. "Consensus control of multi-agent systems with delays." Electronic Research Archive 32, no. 8 (2024): 4887–904. http://dx.doi.org/10.3934/era.2024224.

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<p>This paper concerns the consensus problem of linear time-invariant multi-agent systems (MASs) with multiple state delays and communicate delays. Consensus control is widely applied in spacecraft formation, sensor networks, robotic manipulators, autonomous vehicles, and others. By introducing a linear transformation, the consensus problem of the delayed MAS under an undirected network was converted into a robust asymptotic stability problem associated with the eigenvalues of the normalized Laplacian matrix of the network. By means of the argument principle and optimization technologies

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Ren, Jingyao, Eric Ewing, T. K. Satish Kumar, Sven Koenig, and Nora Ayanian. "Map Connectivity and Empirical Hardness of Grid-based Multi-Agent Pathfinding Problem." Proceedings of the International Conference on Automated Planning and Scheduling 34 (May 30, 2024): 484–88. http://dx.doi.org/10.1609/icaps.v34i1.31508.

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We present an empirical study of the relationship between map connectivity and the empirical hardness of the multi-agent pathfinding (MAPF) problem. By analyzing the second smallest eigenvalue (commonly known as lambda2) of the normalized Laplacian matrix of different maps, our initial study indicates that maps with smaller lambda2 tend to create more challenging instances when agents are generated uniformly randomly. Additionally, we introduce a map generator based on Quality Diversity (QD) that is capable of producing maps with specified lambda2 ranges, offering a possible way for generating

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SUN, YU, JIAHUI ZOU, MEIFENG DAI, XIAOQIAN WANG, HUALONG TANG, and WEIYI SU. "EIGENTIME IDENTITY OF THE WEIGHTED KOCH NETWORKS." Fractals 26, no. 03 (2018): 1850042. http://dx.doi.org/10.1142/s0218348x18500421.

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The eigenvalues of the transition matrix of a weighted network provide information on its structural properties and also on some relevant dynamical aspects, in particular those related to biased walks. Although various dynamical processes have been investigated in weighted networks, analytical research about eigentime identity on such networks is much less. In this paper, we study analytically the scaling of eigentime identity for weight-dependent walk on small-world networks. Firstly, we map the classical Koch fractal to a network, called Koch network. According to the proposed mapping, we pr

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Ficara, Annamaria, Lucia Cavallaro, Francesco Curreri, et al. "Criminal networks analysis in missing data scenarios through graph distances." PLOS ONE 16, no. 8 (2021): e0255067. http://dx.doi.org/10.1371/journal.pone.0255067.

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Data collected in criminal investigations may suffer from issues like: (i) incompleteness, due to the covert nature of criminal organizations; (ii) incorrectness, caused by either unintentional data collection errors or intentional deception by criminals; (iii) inconsistency, when the same information is collected into law enforcement databases multiple times, or in different formats. In this paper we analyze nine real criminal networks of different nature (i.e., Mafia networks, criminal street gangs and terrorist organizations) in order to quantify the impact of incomplete data, and to determ

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Wang, Zhixiao, Zhaotong Chen, Ya Zhao, and Shaoda Chen. "A Community Detection Algorithm Based on Topology Potential and Spectral Clustering." Scientific World Journal 2014 (2014): 1–9. http://dx.doi.org/10.1155/2014/329325.

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Community detection is of great value for complex networks in understanding their inherent law and predicting their behavior. Spectral clustering algorithms have been successfully applied in community detection. This kind of methods has two inadequacies: one is that the input matrixes they used cannot provide sufficient structural information for community detection and the other is that they cannot necessarily derive the proper community number from the ladder distribution of eigenvector elements. In order to solve these problems, this paper puts forward a novel community detection algorithm

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Atay, Fatihcan M. "The consensus problem in networks with transmission delays." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 371, no. 1999 (2013): 20120460. http://dx.doi.org/10.1098/rsta.2012.0460.

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We study discrete- and continuous-time consensus problems on networks in the presence of distributed time delays. We focus on information transmission delays, as opposed to information processing delays, so that each node of the network compares its current state with the past states of its neighbours. We consider directed and weighted networks where the connection structure is described by a normalized Laplacian matrix and show that consensus is achieved if and only if the underlying graph contains a directed spanning tree. This statement holds independently of the transmission delays, which

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Wichmann, David, Christian Kehl, Henk A. Dijkstra, and Erik van Sebille. "Detecting flow features in scarce trajectory data using networks derived from symbolic itineraries: an application to surface drifters in the North Atlantic." Nonlinear Processes in Geophysics 27, no. 4 (2020): 501–18. http://dx.doi.org/10.5194/npg-27-501-2020.

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Abstract. The basin-wide surface transport of tracers such as heat, nutrients and plastic in the North Atlantic Ocean is organized into large-scale flow structures such as the Western Boundary Current and the Subtropical and Subpolar gyres. Being able to identify these features from drifter data is important for studying tracer dispersal but also for detecting changes in the large-scale surface flow due to climate change. We propose a new and conceptually simple method to detect groups of trajectories with similar dynamical behaviour from drifter data using network theory and normalized cut sp

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HILDEBRAND, ROLAND, STEFANO MANCINI, and SIMONE SEVERINI. "Combinatorial laplacians and positivity under partial transpose." Mathematical Structures in Computer Science 18, no. 1 (2008): 205–19. http://dx.doi.org/10.1017/s0960129508006634.

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The density matrices of graphs are combinatorial laplacians normalised to have trace one (Braunstein et al. 2006b). If the vertices of a graph are arranged as an array, its density matrix carries a block structure with respect to which properties such as separability can be considered. We prove that the so-called degree-criterion, which was conjectured to be necessary and sufficient for the separability of density matrices of graphs, is equivalent to the PPT-criterion. As such, it is not sufficient for testing the separability of density matrices of graphs (we provide an explicit example). Non

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Athira, V. S., Vijayakumar S. Muni, Kallu Vetty Muhammed Rafeek, and Gudala Janardhana Reddy. "Controllability of consensus heterogeneous multi-agent networks over continuous time scale." Control and Cybernetics 52, no. 2 (2023): 199–245. http://dx.doi.org/10.2478/candc-2023-0037.

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Abstract The research, presented in this paper, concernes the controllability of a multi-agent network with a directed, unweighted, cooperative, and time-invariant communication topology. The network’s agents follow linear and heterogeneous dynamics, encompassing first-order, second-order, and third-order differential equations over continuous time. Two classes of neighbour-based linear distributed control protocols are considered: the first one utilises average feedback from relative velocities/relative accelerations, and the second one utilises feedback from absolute velocities/absolute acce

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Wang, Jianjia, Chenyue Lin, and Yilei Wang. "Thermodynamic Entropy in Quantum Statistics for Stock Market Networks." Complexity 2019 (April 21, 2019): 1–11. http://dx.doi.org/10.1155/2019/1817248.

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The stock market is a dynamical system composed of intricate relationships between financial entities, such as banks, corporations, and institutions. Such a complex interactive system can be represented by the network structure. The underlying mechanism of stock exchange establishes a time-evolving network among companies and individuals, which characterise the correlations of stock prices in the time sequential trades. Here, we develop a novel technique in quantum statistics to analyse the financial market evolution. We commence from heat bath analogy where the normalised Laplacian matrix pla

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Cai, Min, JiaJia Wang, and Shumin Zhang. "Normalized Laplacian Spectral Ratio of Graphs." Parallel Processing Letters, February 14, 2025. https://doi.org/10.1142/s0129626425300016.

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The normalized Laplacian spectrum of a graph is the eigenvalues of the associated normalized Laplacian matrix. The quotient between the largest and the second smallest normalized Laplacian matrix eigenvalues of a connected graph, is called the normalized Laplacian spectral ratio. Some bounds of the normalized Laplacian spectral ratio of a connected graph are considered. In the paper, we improve a relation of the normalized Laplacian spectral ratio of regular graphs, and we obtain the effects on the normalized Laplacian spectral ratio of a graph by the Operations.

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Wu, Chai Wah. "Multipartite Separability of Laplacian Matrices of Graphs." Electronic Journal of Combinatorics 16, no. 1 (2009). http://dx.doi.org/10.37236/150.

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Recently, Braunstein et al. introduced normalized Laplacian matrices of graphs as density matrices in quantum mechanics and studied the relationships between quantum physical properties and graph theoretical properties of the underlying graphs. We provide further results on the multipartite separability of Laplacian matrices of graphs. In particular, we identify complete bipartite graphs whose normalized Laplacian matrix is multipartite entangled under any vertex labeling. Furthermore, we give conditions on the vertex degrees such that there is a vertex labeling under which the normalized Lapl

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