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Journal articles on the topic 'Laplacian matrix'

Author: Grafiati

Published: 4 June 2021

Last updated: 26 April 2022

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1

Zakiyyah, A. Y. "Some result on integrality of several matrix representation of complete r-uniform hypergraph." Journal of Physics: Conference Series 2157, no. 1 (January 1, 2022): 012006. http://dx.doi.org/10.1088/1742-6596/2157/1/012006.

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Abstract The eigenvalues of matrix representation hypergraph have the possibility that all of it can be an integer or otherwise. The Laplacian integral hypergraphs are those hypergraphs whose Laplacian spectrum consists entirely of integers likewise the definition of signless Laplacian integral and Seidel integral. This research focuses on the properties of the entry matrix and formulates the spectrum of the Laplacian matrix, Signless Laplacian matrix, and Seidel Matrix of complete r-uniform hypergraphs. Hence, it can determine the integrality of the representation matrix respectively. The result concludes that complete r-uniform hypergraphs are Laplacian integral, signless Laplacian integral, and Seidel integral.

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2

Lorenzen, Kate. "Cospectral constructions for several graph matrices using cousin vertices." Special Matrices 10, no. 1 (June 28, 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, signless Laplacian matrix, normalized Laplacian matrix, and distance matrix. We conclude by enumerating the prevelance of this construction in small graphs for the adjacency matrix, combinatorial Laplacian matrix, and distance Laplacian matrix.

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3

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 normalized Laplacian matrix if and only if the signed network is balanced and bipartite. Finally an expression of the coefficient of normalized Laplacian characteristic polynomial is present.

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4

D., Cokilavany. "On Some Bounds of Extended Signless Laplacian Matrix Indices." Journal of Advanced Research in Dynamical and Control Systems 12, SP4 (March 31, 2020): 1816–21. http://dx.doi.org/10.5373/jardcs/v12sp4/20201667.

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5

Wu, Tingzeng, and Tian Zhou. "The Characterizing Properties of (Signless) Laplacian Permanental Polynomials of Almost Complete Graphs." Journal of Mathematics 2021 (September 30, 2021): 1–7. http://dx.doi.org/10.1155/2021/9161508.

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Let G be a graph with n vertices, and let L G and Q G denote the Laplacian matrix and signless Laplacian matrix, respectively. The Laplacian (respectively, signless Laplacian) permanental polynomial of G is defined as the permanent of the characteristic matrix of L G (respectively, Q G ). In this paper, we show that almost complete graphs are determined by their (signless) Laplacian permanental polynomials.

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6

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 (May 22, 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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Trinajstic, Nenad, Darko Babic, Sonja Nikolic, Dejan Plavsic, Dragan Amic, and Zlatko Mihalic. "The Laplacian matrix in chemistry." Journal of Chemical Information and Modeling 34, no. 2 (March 1, 1994): 368–76. http://dx.doi.org/10.1021/ci00018a023.

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8

Tuna, S. Emre. "Synchronization under matrix-weighted Laplacian." Automatica 73 (November 2016): 76–81. http://dx.doi.org/10.1016/j.automatica.2016.06.012.

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9

Ameenal Bibi, K., B. Vijayalakshmi, and R. Jothilakshmi. "Bounds of Laplacian Energy of a Hypercube Graph." International Journal of Engineering & Technology 7, no. 4.10 (October 2, 2018): 582. http://dx.doi.org/10.14419/ijet.v7i4.10.21287.

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Let Qn denote the n – dimensional hypercube with order 2n and size n2n-1. The Laplacian L is defined by L = D where D is the degree matrix and A is the adjacency matrix with zero diagonal entries. The Laplacian is a symmetric positive semidefinite. Let µ1 ≥ µ2 ≥ ....µn-1 ≥ µn = 0 be the eigen values of the Laplacian matrix. The Laplacian energy is defined as LE(G) = . In this paper, we defined Laplacian energy of a Hypercube graph and also attained the lower bounds.

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10

Reinhart, Carolyn. "The normalized distance Laplacian." Special Matrices 9, no. 1 (January 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 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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11

Vrba, Antonín. "The permanent of the Laplacian matrix of a bipartite graph." Czechoslovak Mathematical Journal 36, no. 1 (1986): 7–17. http://dx.doi.org/10.21136/cmj.1986.102059.

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12

Jog, S. R., and Raju Kotambari. "On the Adjacency, Laplacian, and Signless Laplacian Spectrum of Coalescence of Complete Graphs." Journal of Mathematics 2016 (2016): 1–11. http://dx.doi.org/10.1155/2016/5906801.

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Coalescence as one of the operations on a pair of graphs is significant due to its simple form of chromatic polynomial. The adjacency matrix, Laplacian matrix, and signless Laplacian matrix are common matrices usually considered for discussion under spectral graph theory. In this paper, we compute adjacency, Laplacian, and signless Laplacian energy (Qenergy) of coalescence of pair of complete graphs. Also, as an application, we obtain the adjacency energy of subdivision graph and line graph of coalescence from itsQenergy.

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13

DAI, MEIFENG, YONGBO HOU, CHANGXI DAI, TINGTING JU, YU SUN, and WEIYI SU. "CHARACTERISTIC POLYNOMIAL OF ADJACENCY OR LAPLACIAN MATRIX FOR WEIGHTED TREELIKE NETWORKS." Fractals 27, no. 05 (August 2019): 1950074. http://dx.doi.org/10.1142/s0218348x19500749.

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In recent years, weighted networks have been extensively studied in various fields. This paper studies characteristic polynomial of adjacency or Laplacian matrix for weighted treelike networks. First, a class of weighted treelike networks with a weight factor is introduced. Then, the relationships of adjacency or the Laplacian matrix at two successive generations are obtained. Finally, according to the operation of the block matrix, we obtain the analytic expression of the characteristic polynomial of the adjacency or the Laplacian matrix. The obtained results lay the foundation for the future study of adjacency spectrum or Laplacian spectrum.

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14

Barik, S., D. Kalita, S. Pati, and G. Sahoo. "Spectra of Graphs Resulting from Various Graph Operations and Products: a Survey." Special Matrices 6, no. 1 (September 1, 2018): 323–42. http://dx.doi.org/10.1515/spma-2018-0027.

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AbstractLet G be a graph on n vertices and A(G), L(G), and |L|(G) be the adjacency matrix, Laplacian matrix and signless Laplacian matrix of G, respectively. The paper is essentially a survey of known results about the spectra of the adjacency, Laplacian and signless Laplacian matrix of graphs resulting from various graph operations with special emphasis on corona and graph products. In most cases, we have described the eigenvalues of the resulting graphs along with an explicit description of the structure of the corresponding eigenvectors.

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15

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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16

Zhou, Sihang, Xinwang Liu, Jiyuan Liu, Xifeng Guo, Yawei Zhao, En Zhu, Yongping Zhai, Jianping Yin, and Wen Gao. "Multi-View Spectral Clustering with Optimal Neighborhood Laplacian Matrix." Proceedings of the AAAI Conference on Artificial Intelligence 34, no. 04 (April 3, 2020): 6965–72. http://dx.doi.org/10.1609/aaai.v34i04.6180.

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Multi-view spectral clustering aims to group data into different categories by optimally exploring complementary information from multiple Laplacian matrices. However, existing methods usually linearly combine a group of pre-specified first-order Laplacian matrices to construct an optimal Laplacian matrix, which may result in limited representation capability and insufficient information exploitation. In this paper, we propose a novel optimal neighborhood multi-view spectral clustering (ONMSC) algorithm to address these issues. Specifically, the proposed algorithm generates an optimal Laplacian matrix by searching the neighborhood of both the linear combination of the first-order and high-order base Laplacian matrices simultaneously. This design enhances the representative capacity of the optimal Laplacian and better utilizes the hidden high-order connection information, leading to improved clustering performance. An efficient algorithm with proved convergence is designed to solve the resultant optimization problem. Extensive experimental results on 9 datasets demonstrate the superiority of our algorithm against state-of-the-art methods, which verifies the effectiveness and advantages of the proposed ONMSC.

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17

Hu, Xiaoxue, and Grace Kalaso. "Computing the Permanent of the Laplacian Matrices of Nonbipartite Graphs." Journal of Mathematics 2021 (June 22, 2021): 1–4. http://dx.doi.org/10.1155/2021/6621029.

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Let G be a graph with Laplacian matrix L G . Denote by per L G the permanent of L G . In this study, we investigate the problem of computing the permanent of the Laplacian matrix of nonbipartite graphs. We show that the permanent of the Laplacian matrix of some classes of nonbipartite graphs can be formulated as the composite of the determinants of two matrices related to those Laplacian matrices. In addition, some recursion formulas on per L G are deduced.

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18

Alhevaz, Abdollah, Maryam Baghipur, and Somnath Paul. "On the distance signless Laplacian spectral radius and the distance signless Laplacian energy of graphs." Discrete Mathematics, Algorithms and Applications 10, no. 03 (June 2018): 1850035. http://dx.doi.org/10.1142/s1793830918500350.

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The distance signless Laplacian spectral radius of a connected graph [Formula: see text] is the largest eigenvalue of the distance signless Laplacian matrix of [Formula: see text], defined as [Formula: see text], where [Formula: see text] is the distance matrix of [Formula: see text] and [Formula: see text] is the diagonal matrix of vertex transmissions of [Formula: see text]. In this paper, we determine some bounds on the distance signless Laplacian spectral radius of [Formula: see text] based on some graph invariants, and characterize the extremal graphs. In addition, we define distance signless Laplacian energy, similar to that in [J. Yang, L. You and I. Gutman, Bounds on the distance Laplacian energy of graphs, Kragujevac J. Math. 37 (2013) 245–255] and give some bounds on the distance signless Laplacian energy of graphs.

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19

Atik, Fouzul, and Pratima Panigrahi. "On the distance and distance signless Laplacian eigenvalues of graphs and the smallest Gersgorin disc." Electronic Journal of Linear Algebra 34 (February 21, 2018): 191–204. http://dx.doi.org/10.13001/1081-3810.3510.

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The \emph{distance matrix} of a simple connected graph $G$ is $D(G)=(d_{ij})$, where $d_{ij}$ is the distance between the $i$th and $j$th vertices of $G$. The \emph{distance signless Laplacian matrix} of the graph $G$ is $D_Q(G)=D(G)+Tr(G)$, where $Tr(G)$ is a diagonal matrix whose $i$th diagonal entry is the transmission of the vertex $i$ in $G$. In this paper, first, upper and lower bounds for the spectral radius of a nonnegative matrix are constructed. Applying this result, upper and lower bounds for the distance and distance signless Laplacian spectral radius of graphs are given, and the extremal graphs for these bounds are obtained. Also, upper bounds for the modulus of all distance (respectively, distance signless Laplacian) eigenvalues other than the distance (respectively, distance signless Laplacian) spectral radius of graphs are given. These bounds are probably first of their kind as the authors do not find in the literature any bound for these eigenvalues. Finally, for some classes of graphs, it is shown that all distance (respectively, distance signless Laplacian) eigenvalues other than the distance (respectively, distance signless Laplacian) spectral radius lie in the smallest Ger\^sgorin disc of the distance (respectively, distance signless Laplacian) matrix.

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20

Jovanovic, Irena, and Zoran Stanic. "Spectral distances of graphs based on their different matrix representations." Filomat 28, no. 4 (2014): 723–34. http://dx.doi.org/10.2298/fil1404723j.

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The investigation of the spectral distances of graphs that started in [3] (I. Jovanovic, Z. Stanic, Spectral distances of graphs, Linear Algebra Appl., 436 (2012) 1425-1435.) is continued by defining Laplacian and signless Laplacian spectral distances and considering their relations to the spectral distances based on the adjacency matrix of graph. Some separate results concerning the defined distances are given, and the initial spectral distances in certain sets of graphs are investigated. Computational data on Laplacian and signless Laplacian spectral distances are provided.

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21

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 (February 26, 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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22

Silver, Daniel S., and Susan G. Williams. "Knot invariants from Laplacian matrices." Journal of Knot Theory and Its Ramifications 28, no. 09 (August 2019): 1950058. http://dx.doi.org/10.1142/s0218216519500585.

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A checkerboard graph of a special diagram of an oriented link is made a directed, edge-weighted graph in a natural way so that a principal submatrix of its Laplacian matrix is a Seifert matrix of the link. Doubling and weighting the edges of the graph produces a second Laplacian matrix such that a principal submatrix is an Alexander matrix of the link. The Goeritz matrix and signature invariants are obtained in a similar way. A device introduced by Kauffman makes it possible to apply the method to general diagrams.

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23

Chebotarev, Pavel, and Rafig Agaev. "Forest matrices around the Laplacian matrix." Linear Algebra and its Applications 356, no. 1-3 (November 2002): 253–74. http://dx.doi.org/10.1016/s0024-3795(02)00388-9.

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24

Vrba, Antonín. "Principal subpermanents of the Laplacian matrix." Linear and Multilinear Algebra 19, no. 4 (July 1986): 335–46. http://dx.doi.org/10.1080/03081088608817728.

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25

Jahanbani, Akbar, Seyed Mahmoud Sheikholeslami, and Rana Khoeilar. "On the Spectrum of Laplacian Matrix." Mathematical Problems in Engineering 2021 (June 4, 2021): 1–4. http://dx.doi.org/10.1155/2021/8096874.

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Let G be a simple graph of order n . The matrix ℒ G = D G − A G is called the Laplacian matrix of G , where D G and A G denote the diagonal matrix of vertex degrees and the adjacency matrix of G , respectively. Let l 1 G , l n − 1 G be the largest eigenvalue, the second smallest eigenvalue of ℒ G respectively, and λ 1 G be the largest eigenvalue of A G . In this paper, we will present sharp upper and lower bounds for l 1 G and l n − 1 G . Moreover, we investigate the relation between l 1 G and λ 1 G .

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26

Lu, Shi Fang, and Jin Yu Zou. "The New Class of Laplacian Integral Graphs." Advanced Materials Research 989-994 (July 2014): 2643–46. http://dx.doi.org/10.4028/www.scientific.net/amr.989-994.2643.

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Graph is Laplacian integral, if all the eigenvalues of its Laplacian matrix are integral. In this paper, we obtain that the Laplacian characteristic polynomials of graphs by calculation. Characterizes the new class of Laplacian integral graphs .

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27

Sharafdini, R., and A. Z. Abdian. "Signless Laplacian determinations of some graphs with independent edges." Carpathian Mathematical Publications 10, no. 1 (July 3, 2018): 185–96. http://dx.doi.org/10.15330/cmp.10.1.185-196.

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Let $G$ be a simple undirected graph. Then the signless Laplacian matrix of $G$ is defined as $D_G + A_G$ in which $D_G$ and $A_G$ denote the degree matrix and the adjacency matrix of $G$, respectively. The graph $G$ is said to be determined by its signless Laplacian spectrum (DQS, for short), if any graph having the same signless Laplacian spectrum as $G$ is isomorphic to $G$. We show that $G\sqcup rK_2$ is determined by its signless Laplacian spectra under certain conditions, where $r$ and $K_2$ denote a natural number and the complete graph on two vertices, respectively. Applying these results, some DQS graphs with independent edges are obtained.

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Exman, Iaakov, and Rawi Sakhnini. "Linear Software Models: Bipartite Isomorphism between Laplacian Eigenvectors and Modularity Matrix Eigenvectors." International Journal of Software Engineering and Knowledge Engineering 28, no. 07 (July 2018): 897–935. http://dx.doi.org/10.1142/s0218194018400107.

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We have recently shown that one can obtain the numbers and sizes of modules of a software system from the eigenvectors of its modularity matrix symmetrized and weighted by an affinity matrix. However such a weighting still demands a suitable definition of an affinity. This paper offers an alternative way to obtain the same results by means of the eigenvectors of a Laplacian matrix, directly obtained from the modularity matrix without the need of weighting. These two formalizations stand in a mutual isomorphism. We call it bipartite isomorphism since it is most straightforwardly shown by deriving the Laplacian from the modularity matrix and vice versa through the intermediate bipartite graph between two separate sets: the structors’ and the functionals’ sets. This isomorphism is also demonstrated through the equation defining the Laplacian in terms of the modularity matrix, or by the direct mapping of the respective matrices’ eigenvectors. Both matrices and the bipartite graph reflect one central idea: modules are connected components with high cohesion. The Laplacian matrix technique, of which the Fiedler vector is of central importance, is illustrated by case studies. An important claim of this paper is that, independently of the modularity matrix- and Laplacian matrix-specific properties, behind these two alternative matrices there is just one unified algebraic theory of software composition — the Linear Software Models — here concerning the application of the matrices’ eigenvectors to software modularity.

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Alhevaz, A., M. Baghipur, E. Hashemi, and S. Paul. "On the sum of the distance signless Laplacian eigenvalues of a graph and some inequalities involving them." Discrete Mathematics, Algorithms and Applications 12, no. 01 (December 20, 2019): 2050006. http://dx.doi.org/10.1142/s1793830920500068.

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The distance signless Laplacian matrix of a connected graph [Formula: see text] is defined as [Formula: see text], where [Formula: see text] is the distance matrix of [Formula: see text] and [Formula: see text] is the diagonal matrix of vertex transmissions of [Formula: see text]. If [Formula: see text] are the distance signless Laplacian eigenvalues of a simple graph [Formula: see text] of order [Formula: see text] then we put forward the graph invariants [Formula: see text] and [Formula: see text] for the sum of [Formula: see text]-largest and the sum of [Formula: see text]-smallest distance signless Laplacian eigenvalues of a graph [Formula: see text], respectively. We obtain lower bounds for the invariants [Formula: see text] and [Formula: see text]. Then, we present some inequalities between the vertex transmissions, distance eigenvalues, distance Laplacian eigenvalues, and distance signless Laplacian eigenvalues of graphs. Finally, we give some new results and bounds for the distance signless Laplacian energy of graphs.

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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 (October 26, 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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Marques da Silva, Celso, Maria Aguieiras Alvarez de Freitas, and Renata Raposo Del-Vecchio. "A note on a conjecture for the distance Laplacian matrix." Electronic Journal of Linear Algebra 31 (February 5, 2016): 60–68. http://dx.doi.org/10.13001/1081-3810.3002.

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In this note, the graphs of order n having the largest distance Laplacian eigenvalue of multiplicity n â2 are characterized. In particular, it is shown that if the largest eigenvalue of the distance Laplacian matrix of a connected graph G of order n has multiplicity n â 2, then G = S_n or G = K_(p,p), where n = 2p. This resolves a conjecture proposed by M. Aouchiche and P. Hansen in [M. Aouchiche and P. Hansen. A Laplacian for the distance matrix of a graph. Czechoslovak Mathematical Journal, 64(3):751â761, 2014.]. Moreover, it is proved that if G has P_5 as an induced subgraph then the multiplicity of the largest eigenvalue of the distance Laplacian matrix of G is less than n â 3.

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Zhang, Xiaoling, and Jiajia Zhou. "The Distance Laplacian Spectral Radius of Clique Trees." Discrete Dynamics in Nature and Society 2020 (December 9, 2020): 1–8. http://dx.doi.org/10.1155/2020/8855987.

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The distance Laplacian matrix of a connected graph G is defined as ℒ G = Tr G − D G , where D G is the distance matrix of G and Tr G is the diagonal matrix of vertex transmissions of G . The largest eigenvalue of ℒ G is called the distance Laplacian spectral radius of G . In this paper, we determine the graphs with maximum and minimum distance Laplacian spectral radius among all clique trees with n vertices and k cliques. Moreover, we obtain n vertices and k cliques.

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Qi, Linming, Lianying Miao, Weiliang Zhao, and Lu Liu. "A Lower Bound for the Distance Laplacian Spectral Radius of Bipartite Graphs with Given Diameter." Mathematics 10, no. 8 (April 14, 2022): 1301. http://dx.doi.org/10.3390/math10081301.

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Let G be a connected, undirected and simple graph. The distance Laplacian matrix L(G) is defined as L(G)=diag(Tr)−D(G), where D(G) denotes the distance matrix of G and diag(Tr) denotes a diagonal matrix of the vertex transmissions. Denote by ρL(G) the distance Laplacian spectral radius of G. In this paper, we determine a lower bound of the distance Laplacian spectral radius of the n-vertex bipartite graphs with diameter 4. We characterize the extremal graphs attaining this lower bound.

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Diao, Zulong, Xin Wang, Dafang Zhang, Yingru Liu, Kun Xie, and Shaoyao He. "Dynamic Spatial-Temporal Graph Convolutional Neural Networks for Traffic Forecasting." Proceedings of the AAAI Conference on Artificial Intelligence 33 (July 17, 2019): 890–97. http://dx.doi.org/10.1609/aaai.v33i01.3301890.

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Graph convolutional neural networks (GCNN) have become an increasingly active field of research. It models the spatial dependencies of nodes in a graph with a pre-defined Laplacian matrix based on node distances. However, in many application scenarios, spatial dependencies change over time, and the use of fixed Laplacian matrix cannot capture the change. To track the spatial dependencies among traffic data, we propose a dynamic spatio-temporal GCNN for accurate traffic forecasting. The core of our deep learning framework is the finding of the change of Laplacian matrix with a dynamic Laplacian matrix estimator. To enable timely learning with a low complexity, we creatively incorporate tensor decomposition into the deep learning framework, where real-time traffic data are decomposed into a global component that is stable and depends on long-term temporal-spatial traffic relationship and a local component that captures the traffic fluctuations. We propose a novel design to estimate the dynamic Laplacian matrix of the graph with above two components based on our theoretical derivation, and introduce our design basis. The forecasting performance is evaluated with two realtime traffic datasets. Experiment results demonstrate that our network can achieve up to 25% accuracy improvement.

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Zakharov, A. A., A. E. Barinov, and A. L. Zhiznyakov. "RECOGNITION OF HUMAN POSE FROM IMAGES BASED ON GRAPH SPECTRA." ISPRS - International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences XL-5/W6 (May 18, 2015): 9–12. http://dx.doi.org/10.5194/isprsarchives-xl-5-w6-9-2015.

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Recognition of human pose is an actual problem in computer vision. To increase the reliability of the recognition it is proposed to use structured information in the form of graphs. The spectrum of graphs is applied for the comparison of the structures. Image skeletonization is used to construct graphs. Line segments are the nodes of the graph. The end point of line segments are the edges of the graph. The angles between adjacent segments are used to set the weights of the adjacency matrix. The Laplacian matrix is used to generate the spectrum graph. The algorithm consists of the following steps. The graph on the basis of the vectorized image is constructed. The angles between the adjacent segments are calculated. The Laplacian matrix on the basis of the linear graph is calculated. The eigenvalues and eigenvectors of the Laplacian matrix are calculated. The spectral matrix is calculated using its eigenvalues and eigenvectors of the Laplacian matrix. The principal component method is used for the data representation in the space of smaller dimensions. The results of the algorithm are given.

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Baghipur, Maryam, Modjtaba Ghorbani, Hilal A. Ganie, and Yilun Shang. "On the Second-Largest Reciprocal Distance Signless Laplacian Eigenvalue." Mathematics 9, no. 5 (March 2, 2021): 512. http://dx.doi.org/10.3390/math9050512.

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The signless Laplacian reciprocal distance matrix for a simple connected graph G is defined as RQ(G)=diag(RH(G))+RD(G). Here, RD(G) is the Harary matrix (also called reciprocal distance matrix) while diag(RH(G)) represents the diagonal matrix of the total reciprocal distance vertices. In the present work, some upper and lower bounds for the second-largest eigenvalue of the signless Laplacian reciprocal distance matrix of graphs in terms of various graph parameters are investigated. Besides, all graphs attaining these new bounds are characterized. Additionally, it is inferred that among all connected graphs with n vertices, the complete graph Kn and the graph Kn−e obtained from Kn by deleting an edge e have the maximum second-largest signless Laplacian reciprocal distance eigenvalue.

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Nagarajan, D., and A. Rameshkumar. "VARIOUS KINDS OF MATRICES IN CYCLOTOMIC GRAPHS." Advances in Mathematics: Scientific Journal 10, no. 12 (December 10, 2021): 3579–96. http://dx.doi.org/10.37418/amsj.10.12.5.

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The component matrix, Laplacian matrix, Distance matrix, Peripheral distance matrix, Distance Laplacian of the cyclotomic graphs and some properties are found. The D-energy, $D_{p}$-energy, $D^{L}$-energy and some indices of the cyclotomic graphs are determined. For the real symmetric matrices, matrices that attain the maximum $L, L_{s}$ and the minimum S are calculated. The Hausdorff distance and optimal matching distance of the cyclotomic graphs are evaluated.

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Ramane, Harishchandra S., Shaila B. Gudimani, and Sumedha S. Shinde. "Signless Laplacian Polynomial and Characteristic Polynomial of a Graph." Journal of Discrete Mathematics 2013 (January 3, 2013): 1–4. http://dx.doi.org/10.1155/2013/105624.

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The signless Laplacian polynomial of a graph G is the characteristic polynomial of the matrix Q(G)=D(G)+A(G), where D(G) is the diagonal degree matrix and A(G) is the adjacency matrix of G. In this paper we express the signless Laplacian polynomial in terms of the characteristic polynomial of the induced subgraphs, and, for regular graph, the signless Laplacian polynomial is expressed in terms of the derivatives of the characteristic polynomial. Using this we obtain the characteristic polynomial of line graph and subdivision graph in terms of the characteristic polynomial of induced subgraphs.

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Guo, Qiao, Yaoping Hou, and Deqiong Li. "The least Laplacian eigenvalue of the unbalanced unicyclic signed graphs with $k$ pendant vertices." Electronic Journal of Linear Algebra 36, no. 36 (June 18, 2020): 390–99. http://dx.doi.org/10.13001/ela.2020.5077.

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Let $\Gamma=(G,\sigma)$ be a signed graph and $L(\Gamma)=D(G)-A(\Gamma)$ be the Laplacian matrix of $\Gamma$, where $D(G)$ is the diagonal matrix of vertex degrees of the underlying graph $G$ and $A(\Gamma)$ is the adjacency matrix of $\Gamma$. It is well-known that the least Laplacian eigenvalue $\lambda_n$ is positive if and only if $\Gamma$ is unbalanced. In this paper, the unique signed graph (up to switching equivalence) which minimizes the least Laplacian eigenvalue among unbalanced connected signed unicyclic graphs with $n$ vertices and $k$ pendant vertices is characterized.

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Ganesh, Swetha, and Sumit Mohanty. "Trees with matrix weights: Laplacian matrix and characteristic-like vertices." Linear Algebra and its Applications 646 (August 2022): 195–237. http://dx.doi.org/10.1016/j.laa.2022.03.029.

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Hou, Yaoping, and Dijian Wang. "Laplacian integral subcubic signed graphs." Electronic Journal of Linear Algebra 37 (February 26, 2021): 163–76. http://dx.doi.org/10.13001/ela.2021.5699.

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A (signed) graph is called Laplacian integral if all eigenvalues of its Laplacian matrix are integers. In this paper, we determine all connected Laplacian integral signed graphs of maximum degree 3; among these signed graphs,there are two classes of Laplacian integral signed graphs, one contains 4 infinite families of signed graphs and another contains 29 individual signed graphs.

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Qi, Xingqin, Edgar Fuller, Rong Luo, Guodong Guo, and Cunquan Zhang. "Laplacian Energy of Digraphs and a Minimum Laplacian Energy Algorithm." International Journal of Foundations of Computer Science 26, no. 03 (April 2015): 367–80. http://dx.doi.org/10.1142/s0129054115500203.

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In spectral graph theory, the Laplacian energy of undirected graphs has been studied extensively. However, there has been little work yet for digraphs. Recently, Perera and Mizoguchi (2010) introduced the directed Laplacian matrix [Formula: see text] and directed Laplacian energy [Formula: see text] using the second spectral moment of [Formula: see text] for a digraph [Formula: see text] with [Formula: see text] vertices, where [Formula: see text] is the diagonal out-degree matrix, and [Formula: see text] with [Formula: see text] whenever there is an arc [Formula: see text] from the vertex [Formula: see text] to the vertex [Formula: see text] and 0 otherwise. They studied the directed Laplacian energies of two special families of digraphs (simple digraphs and symmetric digraphs). In this paper, we extend the study of Laplacian energy for digraphs which allow both simple and symmetric arcs. We present lower and upper bounds for the Laplacian energy for such digraphs and also characterize the extremal graphs that attain the lower and upper bounds. We also present a polynomial algorithm to find an optimal orientation of a simple undirected graph such that the resulting oriented graph has the minimum Laplacian energy among all orientations. This solves an open problem proposed by Perera and Mizoguchi at 2010.

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Catral, Minerva, Lorenzo Ciardo, Leslie Hogben, and Carolyn Reinhart. "Spectra of products of digraphs." Electronic Journal of Linear Algebra 36, no. 36 (December 1, 2020): 744–63. http://dx.doi.org/10.13001/ela.2020.5243.

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A unified approach to the determination of eigenvalues and eigenvectors of specific matrices associated with directed graphs is presented. Matrices studied include the new distance matrix, with natural extensions to the distance Laplacian and distance signless Laplacian, in addition to the new adjacency matrix, with natural extensions to the Laplacian and signless Laplacian. Various sums of Kronecker products of nonnegative matrices are introduced to model the Cartesian and lexicographic products of digraphs. The Jordan canonical form is applied extensively to the analysis of spectra and eigenvectors. The analysis shows that Cartesian products provide a method for building infinite families of transmission regular digraphs with few distinct distance eigenvalues.

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YANG, YUXIANG, and ZENGFU WANG. "AN APPLICATION OF MATTING LAPLACIAN MATRIX TO RANGE IMAGE SUPER-RESOLUTION." International Journal of Information Acquisition 08, no. 04 (December 2011): 273–80. http://dx.doi.org/10.1142/s0219878911002513.

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This paper describes a successful application of Matting Laplacian Matrix to the problem of generating high-resolution range images. The Matting Laplacian Matrix in this paper exploits the fact that discontinuities in range and coloring tend to co-align, which enables us to generate high-resolution range image by integrating regular camera image into the range data. Using one registered and potentially high-resolution camera image as reference, we iteratively refine the input low-resolution range image, in terms of both spatial resolution and depth precision. We show that by using such a Matting Laplacian Matrix, we can get high-quality high-resolution range images.

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Hu, Rongyao, Leyuan Zhang, and Jian Wei. "Adaptive Laplacian Support Vector Machine for Semi-supervised Learning." Computer Journal 64, no. 7 (April 30, 2021): 1005–15. http://dx.doi.org/10.1093/comjnl/bxab024.

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Abstract Laplacian support vector machine (LapSVM) is an extremely popular classification method and relies on a small number of labels and a Laplacian regularization to complete the training of the support vector machine (SVM). However, the training of SVM model and Laplacian matrix construction are usually two independent process. Therefore, In this paper, we propose a new adaptive LapSVM method to realize semi-supervised learning with a primal solution. Specifically, the hinge loss of unlabelled data is considered to maximize the distance between unlabelled samples from different classes and the process of dealing with labelled data are similar to other LapSVM methods. Besides, the proposed method embeds the Laplacian matrix acquisition into the SVM training process to improve the effectiveness of Laplacian matrix and the accuracy of new SVM model. Moreover, a novel optimization algorithm considering primal solver is proposed to our adaptive LapSVM model. Experimental results showed that our method outperformed all comparison methods in terms of different evaluation metrics on both real datasets and synthetic datasets.

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Cvetkovic, Dragos, and Slobodan Simic. "Towards a spectral theory of graphs based on the signless Laplacian, I." Publications de l'Institut Math?matique (Belgrade) 85, no. 99 (2009): 19–33. http://dx.doi.org/10.2298/pim0999019c.

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A spectral graph theory is a theory in which graphs are studied by means of eigenvalues of a matrix M which is in a prescribed way defined for any graph. This theory is called M-theory. We outline a spectral theory of graphs based on the signless Laplacians Q and compare it with other spectral theories, in particular with those based on the adjacency matrix A and the Laplacian L. The Q-theory can be composed using various connections to other theories: equivalency with A-theory and L-theory for regular graphs, or with L-theory for bipartite graphs, general analogies with A-theory and analogies with A-theory via line graphs and subdivision graphs. We present results on graph operations, inequalities for eigenvalues and reconstruction problems.

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Ramesh, Obbu, and S. Sharief Basha. "An Intuitionistic Fuzzy Graph’s Signless Laplacian Energy." International Journal of Engineering & Technology 7, no. 4.10 (October 2, 2018): 892. http://dx.doi.org/10.14419/ijet.v7i4.10.26782.

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We are extending concept into the Intuitionistic fuzzy graph’ Signless Laplacian energy instead of the Signless Laplacian energy of fuzzy graph. Now we demarcated an Intuitionistic fuzzy graph’s Signless adjacency matrix and also an Intuitionistic fuzzy graph’s Signless Laplacian energy. Here we find the Signless Laplacian energy ‘s Intuitionistic fuzzy graphs above and below boundaries of an with suitable examples.

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Li, Wenjing. "Sparse Manifold Learning based on Laplacian Matrix." Journal of Physics: Conference Series 1187, no. 4 (April 2019): 042022. http://dx.doi.org/10.1088/1742-6596/1187/4/042022.

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Bapat, R. B., S. J. Kirkland, and S. Pati. "The perturbed laplacian matrix of a graph." Linear and Multilinear Algebra 49, no. 3 (December 2001): 219–42. http://dx.doi.org/10.1080/03081080108818697.

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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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