• 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 the Korean Mathematical Society
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• v.43 no.6
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• pp.1269-1287
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• 2006

As a continuation of computing the zeta function of a regular covering graph by Mizuno and Sato in [9], we derive in this paper computational formulae for the zeta functions of a graph bundle and of any (regular or irregular) covering of a graph. If the voltages to derive them lie in an abelian or dihedral group and its fibre is a regular graph, those formulae can be simplified. As a by-product, the zeta function of the cartesian product of a graph and a regular graph is obtained. The same work is also done for a discrete torus and for a discrete Klein bottle.

• Journal of applied mathematics & informatics
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• v.22 no.1_2
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• pp.39-48
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• 2006

Let ${\gamma}$(G) be the domination number of a graph G. It is shown that for any $k {\ge} 0$ there exists a Cartesian graph bundle $B{\Box}_{\varphi}F$ such that ${\gamma}(B{\Box}_{\varphi}F) ={\gamma}(B){\gamma}(F)-2k$. The domination numbers of Cartesian bundles of two cycles are determined exactly when the fibre graph is a triangle or a square. A statement similar to Vizing's conjecture on strong graph bundles is shown not to be true by proving the inequality ${\gamma}(B{\bigotimes}_{\varphi}F){\le}{\gamma}(B){\gamma}(F)$ for strong graph bundles. Examples of graphs Band F with ${\gamma}(B{\bigotimes}_{\varphi}F) < {\gamma}(B){\gamma}(F)$ are given.

• Communications of the Korean Mathematical Society
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• v.10 no.4
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• pp.831-837
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• 1995

In this paper, we introduce a permutation graph over a graph G as a generalization of both a graph bundle over G and a standard permutation graph, and study a characterization of a natural isomorphism and an automorphism of permutation graphs over a graph.

• Honam Mathematical Journal
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• v.27 no.4
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• pp.551-559
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• 2005

A permutation graph over a graph G is a generalization of both a graph bundle and a graph covering over G. In this paper, we characterize the F-permutation graphs over a graph whose chromatic numbers are 2. We determine the chromatic numbers of $C_n$-permutation graphs over a tree and the $K_m$-permutation graphs over a cycle.

• Journal of Institute of Control, Robotics and Systems
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• v.20 no.3
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• pp.372-379
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• 2014

This paper surveys past and state-of-the-art SLAM technologies. The standard methods for solving the SLAM problem are the Kalman filter, particle filter, graph, and bundle adjustment-based methods. Kalman filters such as EKF (Extended Kalman Filter) and UKF (Unscented Kalman Filter) have provided successful results for estimating the state of nonlinear systems and integrating various sensor information. However, traditional EKF-based methods suffer from the increase of computation burden as the number of features increases. To cope with this problem, particle filter-based SLAM approaches such as FastSLAM have been widely used. While particle filter-based methods can deal with a large number of features, the computation time still increases as the map grows. Graph-based SLAM methods have recently received considerable attention, and they can provide successful real-time SLAM results in large urban environments.

• Bulletin of the Korean Mathematical Society
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• v.33 no.1
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• pp.75-86
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• 1996

Let G be a finite simple connected graph with vertex set V(G) and edge set E(G). Let A(G) be the adjacency matrix of G. The characteristic polynomial of G is the characteristic polynomial $\Phi(G;\lambda) = det(\lambda I - A(G))$ of A(G). A zero of $\Phi(G;\lambda)$ is called an eigenvalue of G.

• Journal of the Korean Mathematical Society
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• v.54 no.1
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• pp.227-248
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• 2017

In this paper, we reduce the classification problem of equivariant (topological complex) vector bundles over a simple graph to the classification problem of their isotropy representations at vertices and midpoints of edges. Then, we solve the reduced problem in the case when the simple graph is homeomorphic to a circle. So, the paper could be considered as a generalization of [3].

• Journal of Institute of Control, Robotics and Systems
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• v.18 no.3
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• pp.218-223
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• 2012

Maintenance and improvement of the graph connectivity is very important for decentralized multi-agent systems. Although the CBBA (Consensus-Based Bundle Algorithm) guarantees suboptimal performance and bounded convergence time, it is only valid for connected graphs. In this study, we apply a decentralized estimation procedure that allows each agent to track the algebraic connectivity of a time-varying graph. Based on this estimation, we design a decentralized gradient controller to maintain the graph connectivity while agents are traveling to perform assigned tasks. Simulation result for fully-actuated first-order agents that move in a 2-D plane are presented.

• The Journal of Korea Robotics Society
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• v.12 no.4
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• pp.385-394
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• 2017

Side scanning sonar (SSS) provides valuable information for robot navigation. However using the side scanning sonar images in the navigation was not fully studied. In this paper, we use range data, and side scanning sonar images from UnderWater Simulator (UWSim) and propose measurement models in a feature based simultaneous localization and mapping (SLAM) framework. The range data is obtained by echosounder and sidescanning sonar images from side scan sonar module for UWSim. For the feature, we used the A-KAZE feature for the SSS image matching and adjusting the relative robot pose by SSS bundle adjustment (BA) with Ceres solver. We use BA for the loop closure constraint of pose-graph SLAM. We used the Incremental Smoothing and Mapping (iSAM) to optimize the graph. The optimized trajectory was compared against the dead reckoning (DR).