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

• Honam Mathematical Journal
• /
• v.41 no.1
• /
• pp.67-87
• /
• 2019

The main aim of this study is to characterize new classes of multicone graphs which are determined by their adjacency spectra, their Laplacian spectra, their complement with respect to signless Laplacian spectra and their complement with respect to their adjacency spectra. A multicone graph is defined to be the join of a clique and a regular graph. If n is a positive integer, a friendship graph $F_n$ consists of n edge-disjoint triangles that all of them meet in one vertex. It is proved that any connected graph cospectral to a multicone graph $F_n{\nabla}F_n=K_2{\nabla}nK_2{\nabla}nK_2$ is determined by its adjacency spectra as well as its Laplacian spectra. In addition, we show that if $n{\neq}2$, the complement of these graphs are determined by their adjacency spectra. At the end of the paper, it is proved that multicone graphs $F_n{\nabla}F_n=K_2{\nabla}nK_2{\nabla}nK_2$ are determined by their signless Laplacian spectra and also we prove that any graph cospectral to one of multicone graphs $F_n{\nabla}F_n$ is perfect.

• Korean Journal of Mathematics
• /
• v.29 no.1
• /
• pp.13-24
• /
• 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 applied mathematics & informatics
• /
• v.32 no.1_2
• /
• pp.227-245
• /
• 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.

• Kyungpook Mathematical Journal
• /
• v.49 no.1
• /
• pp.123-131
• /
• 2009

In this paper, we study the signless Laplacian spectral radius of unicyclic graphs with prescribed number of pendant vertices or independence number. We also characterize the extremal graphs completely.

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

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

• Proceedings of the Korean Information Science Society Conference
• /
• 2012.06b
• /
• pp.522-524
• /
• 2012

For exact classification of the defect, good feature selection and classifier is necessary. In this paper, various features such as brightness features, shape features and statistical features are stated and Bayes classifier using Gaussian mixture model is used as classifier. Also feature extraction method based on spectral graph theory is presented. Experimental result shows that feature extraction method using graph Laplacian result in better performance than the result using PCA.

• Journal of Advanced Marine Engineering and Technology
• /
• v.39 no.2
• /
• pp.159-165
• /
• 2015

Motion control schemes are generally classified into three categories (point stabilization, trajectory tracking, and path following). This paper deals with the problem which is associated with the initial deployment of a group of Unmanned Surface Vehicle (USVs) and corresponding point stabilization. To keep the formation of a group of USVs, it is necessary to set the relationship between each vehicle. A forcing functions such as potential fields are designed to keep the formation and a graph Laplacian is used to represent the connectivity between vehicle. In case of fixed topology of the graph representing the communication between the vehicles, the graph Laplacian is assumed constant. However the graph topologies are allowed to change as the vehicles move, and the system dynamics become discontinuous in nature because the graph Laplacian changes as time passes. To check the stability in the stage of deployment, the system is modeled with Kronecker algebra notation. Filippov's calculus of differential equations with discontinuous right hand sides is then used to formally characterize the behavior of USVs. The stability of the system is analyzed with Lyapunov's stability theory and LaSalle's invariance principle, and the validity is shown by checking the variation of state norm.

• Kyungpook Mathematical Journal
• /
• v.50 no.1
• /
• pp.109-116
• /
• 2010

In this paper, we study the signless Laplacian spectral radius of bicyclic graphs with given number of pendant vertices and characterize the extremal graphs.