• Kyungpook Mathematical Journal
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• v.61 no.1
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• pp.155-168
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• 2021

We express the Seidel Laplacian polynomial and Seidel signless Laplacian polynomial of a graph in terms of the Seidel polynomials of induced subgraphs. Further, we determine the Seidel Laplacian polynomial and Seidel signless Laplacian polynomial of the join of regular graphs.

• 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 IKEEE
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• v.4 no.2 s.7
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• pp.288-294
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• 2000

In this paper, an efficient detail extraction technique for a progressive coding scheme is proposed. The existing technique using the top-hat transformation yields an efficient extraction scheme for isolated and visually important details, but yields an inefficient results containing significant redundancy extracting the contour information. The proposed technique using the strong edge feature extraction property of the morphological Laplacian in this paper can reduce the redundancy, and thus provides lower bit-rate. Experimental results show that the proposed technique is more efficient than the existing one, and promise the applicability of the morphological Laplacian operator.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.11 no.4
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• pp.89-99
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• 2007

In this paper, we define the upper and lower solutions for a p-Laplacian system with singular nonlinearity at the boundaries. And we prove the theorem for the upper and power solutions method.

• Korean Journal of Mathematics
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• v.7 no.2
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• pp.139-150
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• 1999

In this paper we establish some bounds for solutions of parabolic one dimensional $p$-Laplacian equation.

• Journal of Korea Multimedia Society
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• v.2 no.3
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• pp.320-328
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• 1999

The Laplacian operator is usually used as a regularization operator which may be used as any differential operator in the regularization iterative restoration. In this paper, several kinds of differential operator and 1-H operator that has been used in our lab as well, as a regularization operator, were compared with each other. In the restoration of noisy motion-blurred images, 1-H operator worked better than Laplacian operator in flat region, but in the edge the Laplacian operator operated better. For noisy gaussian-blurred image, 1-H operator worked better in the edge, while in flat region the Laplacian operator resulted better. In regularization, smoothing the noise and resorting the edges should be considered at the same time, so the regions divided into the flat, the middle, and the detailed, which were processed in separate and compared their MSE. Laplacian and 1-H operator showed to be suitable as the regularization operator, while the other differential operators appeared to be diverged as iterations proceeded.

• Journal of applied mathematics & informatics
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• v.32 no.1_2
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• pp.227-245
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• 2014

Suppose G is a simple graph. The ${\ell}$-eigenvalues ${\delta}_1$, ${\delta}_2$,..., ${\delta}_n$ of G are the eigenvalues of its normalized Laplacian ${\ell}$. The normalized Laplacian Estrada index of the graph G is dened as ${\ell}EE$ = ${\ell}EE$(G) = ${\sum}^n_{i=1}e^{{\delta}_i}$. In this paper the basic properties of ${\ell}EE$ are investigated. Moreover, some lower and upper bounds for the normalized Laplacian Estrada index in terms of the number of vertices, edges and the Randic index are obtained. In addition, some relations between ${\ell}EE$ and graph energy $E_{\ell}$(G) are presented.

• Journal of the Korean Mathematical Society
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• v.45 no.6
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• pp.1785-1801
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• 2008

Noncommutative extensions of the Gross and Beltrami Laplacians, called the quantum Gross Laplacian and the quantum Beltrami Laplacian, resp., are introduced and their basic properties are studied. As noncommutative extensions of the Fourier-Gauss and Fourier-Mehler transforms, we introduce the quantum Fourier-Gauss and quantum Fourier- Mehler transforms. The infinitesimal generators of all differentiable one parameter groups induced by the quantum Fourier-Gauss transform are linear combinations of the quantum Gross Laplacian and quantum Beltrami Laplacian. A characterization of the quantum Fourier-Mehler transform is studied.

• 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.

• Communications of the Korean Mathematical Society
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• v.14 no.1
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• pp.81-97
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• 1999

A sufficient condition for pseudo-Laplacian equations involving critical Sobolev exponents to have positive solutions is established.