• Communications of the Korean Mathematical Society
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• v.11 no.4
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• pp.1159-1174
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• 1996

The spectrum of the Laplacian matrix of a graph gives an information of the structure of the graph. For example, the product of non-zero eigenvalues of the characteristic polynomial of the Laplacian matrix of a graph with n vertices is n times of the number of spanning trees of that graph. The characteristic polynomial of the Laplacian matrix of a graph tells us the number of spanning trees and the connectivity of given graph. in this paper, we compute the characteristic polynomial of the Laplacian matrix of a graph bundle when its voltage lie in an abelian subgroup of the full automorphism group of the fibre; in particular, the automorphism group of the fibre is abelian. Also we study a relation between the characteristic polynomial of the Laplacian matrix of a graph G and that of the Laplacian matrix of a graph bundle over G. Some applications are also discussed.

• Journal of Institute of Control, Robotics and Systems
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• v.20 no.12
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• pp.1225-1230
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• 2014

This paper investigates the group average-consensus and group formation-consensus problems for first-order multi-agent systems. The control protocol for group consensus is designed by considering the positive adjacency elements. Since each intra-group Laplacian matrix cannot be satisfied with the in-degree balance because of the positive adjacency elements between groups, we decompose the Laplacian matrix into an intra-group Laplacian matrix and an inter-group Laplacian matrix. Moreover, average matrices are used in the control protocol to analyze the stability of multi-agent systems with a fixed and undirected communication topology. Using the graph theory and the Lyapunov functional, stability analysis is performed for group average-consensus and group formation-consensus, respectively. Finally, some simulation results are presented to validate the effectiveness of the proposed control protocol for group consensus.

• Journal of Applied and Pure Mathematics
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• v.5 no.3_4
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• pp.255-263
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• 2023

In general, the rigidity problem of braced rectangular frameworks is determined by the connectivity of the bipartite graph induced by given rectangular framework. In this paper, we study how to solve the rigidity problem of the braced rectangular framework using the Laplacian matrix of the matrix induced by a braced rectangular framework.

• Korean Journal of Mathematics
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• v.29 no.1
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• pp.13-24
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• 2021

For a finite commutative ring R with identity 1 ≠ 0, the zero divisor graph ��(R) is a simple connected graph having vertex set as the set of nonzero zero divisors of R, where two vertices x and y are adjacent if and only if xy = 0. We find the signless Laplacian spectrum of the zero divisor graphs ��(ℤn) for various values of n. Also, we find signless Laplacian spectrum of ��(ℤn) for n = pz, z ≥ 2, in terms of signless Laplacian spectrum of its components and zeros of the characteristic polynomial of an auxiliary matrix. Further, we characterise n for which zero divisor graph ��(ℤn) are signless Laplacian integral.

• Journal of Korea Water Resources Association
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• v.32 no.4
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• pp.437-445
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• 1999

The present paper describes the problem of hydrologic response estimation using non-parametric ridge regression method. The method adapted in this work is based on the minimization of the $C_L$ statistics, which is an estimate of the mean square prediction error. For this method, effects of using both the identity matrix and the Laplacian matrix were considered. In addition, we evaluated methods for estimating the error variance of the impulse response. As a result of analyzing synthetic and real data, a good estimation was made when the Laplacian matrix for the weighting matrix and the bias corrected estimate for the error variance were used. The method and procedure presented in present paper will play a robust and effective role on separating hydrologic response.

• 한국전산유체공학회:학술대회논문집
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• 2009.11a
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• pp.44-48
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• 2009

In this study, a vortex visualization method based on the vorticity magnitude is developed. One of the simplest models for a vortex is a vortex filament with the maximum vorticity on its center. The proposed method is based on the observation of this ideal distribution of vorticity magnitude. Laplacian and Hessian matrix of vorticity magnitude are tested for detecting the local maximum of vorticity magnitude. These ideas were applied to wake flow past a sphere. It was found that the Laplacian method is not able to distinguish vortices from the underlying shear layer clearly, while the Hessian matrix method does not suffer from this problem.

• Proceedings of the Korean Vacuum Society Conference
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• 2010.02a
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• pp.472-472
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• 2010

The tomography has played a key role in tokamak plasma diagnostics for image reconstruction. The Phillips-Tikhonov (P-T) regularization method was attempted in this work to reconstruct cross-sectional phantom images of the plasma by minimizing the gradient between adjacent pixel data. Recent studies about the comparison of the several tomographic reconstruction methods showed that the P-T method produced more accurate results. We have studied existing Laplacian matrix used in Phillips-Tikhonov regularization method and developed modified Laplacian matrix (Modified L). The comparison of the reconstruction result by the modified L and existing L showed that modified L produced more accurate result. The difference was significantly pronounced when a portion of plasma was reconstructed. These results can be utilized in the Edge Plasma diagnostics; especially in divertor diagnostics on tokamak a large impact is expected. In addition, accurate reconstruction results from received data in only one direction were confirmed through phantom test by using P-T method with modified L. These results can be applied to the tangentially viewing pin-hole camera diagnostics on tokamak.

• Proceedings of the Korean Society for Bioinformatics Conference
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• 2004.11a
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• pp.265-271
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• 2004

The interaction network of protein -protein plays an important role to understand the various biological functions of cells. Currently, the high -throughput experimental techniques (two -dimensional gel electrophoresis, mass spectroscopy, yeast two -hybrid assay) provide us with the vast amount of data for protein-protein interaction at the proteome scale. In order to recognize the role of each protein in their network, the efficient bioinformatical and computational analysis methods are required. We propose a systematic and mathematical method which can analyze the protein -protein interaction network rigorously and enable us to capture the biological and physical essence of a topological character and stability of protein -protein network, and sensitivity of each protein along the biological pathway of their network. We set up a Laplacian matrix of spectral graph theory based on the protein-protein network of yeast proteome, and perform an eigenvalue analysis and apply a perturbation method on a Laplacian matrix, which result in recognizing the center of protein cluster, the identity of hub proteins around it and their relative sensitivities. Identifying the topology of protein -protein network via a Laplacian matrix, we can recognize the important relation between the biological pathway of yeast proteome and the formalism of master equation. The results of our systematic and mathematical analysis agree well with the experimental findings of yeast proteome. The biological function and meaning of each protein cluster can be explained easily. Our rigorous analysis method is robust for understanding various kinds of networks whether they are biological, social, economical...etc

• Journal of applied mathematics & informatics
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• v.42 no.4
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• pp.725-737
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• 2024

This article introduces the concepts of Energy and Laplacian Energy (LE) of Domination in Hesitancy fuzzy graph (DHFG). Also, the adjacency matrix of a DHFG is defined and proposed the definition of the energy of domination in hesitancy fuzzy graph, and Laplacian energy of domination in hesitancy fuzzy graph is given.

• Korean Journal of Mathematics
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• v.12 no.2
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• pp.201-209
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• 2004

We define the incidence matrices of oriented and nonoriented ${\kappa}$-hypergraphs, respectively. We discuss the ranks of some circulant matrices and show that the rank of the incidence matrices of oriented and nonoriented ${\kappa}$-hypergraphs H are $n$ under a certain condition on the ${\kappa}$-edge set or ${\kappa}$-arc set of H.