• Journal of applied mathematics & informatics
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• v.35 no.1_2
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• pp.83-93
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• 2017

The Cayley graph ${\Gamma}=Cay(G,S)$ is called normal edge-transitive if $N_A(R(G))$ acts transitively on the set of edges of ${\Gamma}$, where $A=Aut({\Gamma})$ and R(G) is the regular subgroup of A. In this paper, we determine all hexavalent normal edge-transitive Cayley graphs on groups of order pqr, where p > q > r > 2 are prime numbers.

• Journal for History of Mathematics
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• v.19 no.2
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• pp.125-140
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• 2006

In 1933, I. Schur introduced a Schur ring in connection with permutation group and regular subgroup. After then, it was studied mostly for purely group theoretical purposes. In 1970s, Klin and Poschel initiated its usage in the investigation of graphs, especially for Cayley and circulant graphs. Nowadays it is known that Schur ring is one of the best way to enumerate Cayley graphs. In this paper we study the origin of Schur ring back to 1933 and keep trace its evolution to graph theory and combinatorics.

• Journal of the Korea Institute of Information and Communication Engineering
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• v.15 no.9
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• pp.1939-1946
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• 2011

Expander graphs are useful in the design and analysis of communication networks. Mukhopadhyay et. al introduced a method to generate a family of expander graphs based on nongroup two predecessor single attractor CA(Cellular Automata). In this paper we propose a method to generate a family of expander graphs based on 60/102 Null boundary CA(NBCA) which is a group CA. The spectral gap generated by our method is larger than that of Mukhopadhyay et. al [12]. As an application we give an algorithm which generate one-way functions whose security lies on the combinatorial properties of our expander graphs. the one-way function using d-regular graph generated by the 60/102 NBCA is based on the Goldreich's construction [5].

• Journal of the Korean Mathematical Society
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• v.49 no.5
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• pp.1031-1051
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• 2012

In this paper, we will investigate the concept of the torsion-graph of an R-module M, in which the set $T(M)^*$ makes up the vertices of the corresponding torsion graph, ${\Gamma}(M)$, with any two distinct vertices forming an edge if $[x:M][y:M]M=0$. We prove that, if ${\Gamma}(M)$ contains a cycle, then $gr({\Gamma}(M)){\leq}4$ and ${\Gamma}(M)$ has a connected induced subgraph ${\overline{\Gamma}}(M)$ with vertex set $\{m{\in}T(M)^*{\mid}Ann(m)M{\neq}0\}$ and diam$({\overline{\Gamma}}(M)){\leq}3$. Moreover, if M is a multiplication R-module, then ${\overline{\Gamma}}(M)$ is a maximal connected subgraph of ${\Gamma}(M)$. Also ${\overline{\Gamma}}(M)$ and ${\overline{\Gamma}}(S^{-1}M)$ are isomorphic graphs, where $S=R{\backslash}Z(M)$. Furthermore, we show that, if ${\overline{\Gamma}}(M)$ is uniquely complemented, then $S^{-1}M$ is a von Neumann regular module or ${\overline{\Gamma}}(M)$ is a star graph.

• Journal of the Korea Institute of Information and Communication Engineering
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• v.19 no.12
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• pp.2865-2870
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• 2015

Recently, due to the hardware computing enhancement such as quantum computers, the amount of information that can be processed in a short period of time is growing exponentially. The cryptography system proposed by Koblitz and Fellows has a problem that it can not be guaranteed that the problem finding perfect dominating set is NP-complete in specific 3-regular graphs because the number of invariant polynomial can not be generated enough. In this paper, we propose an enhanced method to improve the vulnerability in 3-regular graph by generating plenty of invariant polynomials.

• The Journal of Korean Institute of Communications and Information Sciences
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• v.29 no.10C
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• pp.1363-1369
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• 2004

LDPC(Low Density Parity Check) codes are described by bipartite graphs with bit nodes and parity-check nodes. Tanner derived minimum distance bounds of the regular LDPC code in terms of the eigenvalues of the associated adjacency matrix. In this paper we generalize the Tanner's results. We derive minimum distance bounds applicable to both regular and blockwise-irregular LDPC codes. The first bound considers the relation between bit nodes in a minimum-weight codeword, and the second one considers the connectivity between parity nodes adjacent to a minimum-weight codeword. The derived bounds make it possible to describe the distance property of the code in terms of the eigenvalues of the associated matrix.

• Journal of Korean Elementary Science Education
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• v.31 no.3
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• pp.321-333
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• 2012

The purpose of this study was to examine elementary school students' characteristics and difficulties in drawing and interpreting a line graph, and to present educational implications. Twenty five students(4th grader: 6, 5th grader: 9, and 6th grader: 10) at an elementary school participated in this study. We used a student's task which was about graphing on a given data table and interpreting his/her graph. The data table was on heating 200mL and 500mL of water and measuring their temperature at regular time intervals. We collected multiple source of data, and data analyzed based on the sub-variables of TOGS. The some results of this study are as follows: First, five children (20.0%), especially two of 10 sixth graders (20.0%), could not construct a line graph about a given data table. Second, twenty students (80.0%) had the ability on 'Scaling axes' and on 'Assigning variables to the axes', however, only a student understood why the time is on the longitudinal axis and the temperature is on the vertical axis. Third, in the case of 'Plotting points', twelve children (48.0%) could drew two graphs on a coordinate. Fourth, in the case of 'Selecting the corresponding value for Y (or X)', twenty student had little difficulty. on 'Describing the relationship between variables', seventeen students (68.0%) understood the relationship between time and temperature of water, and the relationship between temperature and amount of water. Finally, eleven students (44%) had the ability on 'Interrelating and extrapolation graphs.' Educational implications are also presented in this paper.

• Journal of the Korea Society of Computer and Information
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• v.20 no.1
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• pp.227-235
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• 2015

In this paper, I propose a linear time algorithm devised to produce exact solution to NP-complete maximum clique problem. The proposed algorithm firstly, from a given graph G=(V,E), sets vertex $v_i$ of the maximum degree ${\Delta}(G)$ as clique's major vertex. It then selects vertex $v_j$ of ${\Delta}(G)$ among vertices $N_G(v_i)$ that are adjacent to $v_i$, only to determine $N_G(v_i){\cap}N_G(v_j)$ as candidate cliques w and $v_k$. Next it obtains $w=w{\cap}N_G(v_k)$ by sorting $d_G(v_k)$ in the descending order. Lastly, the algorithm executes the same procedure on $G{\backslash}w$ graph to compare newly attained cliques to previously attained cliques so as to choose the lower. With this simple method, multiple independent cliques would also be attainable. When applied to various regular and irregular graphs, the algorithm proposed in this paper has obtained exact solutions to all the given graphs linear time O(n).

• Proceedings of the IEEK Conference
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• 2002.07a
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• pp.301-304
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• 2002

Recently, database technologies have been used commonly. But, ordinary technologies aren't suitable to construct a complicated database such as a classical literature database or an archaeological relic's database. Because this kinds of data are semi-structured data that doesn't have regular structures, database schema can't be defined before databases. We have proposed DREAM model for semi-structured databases. In this model, a database consists of five elements and the model has operations similar to operation of set theory. And further we have introduced dynamic schema "shape" showing structure of each element. We have already realized a prototype of DBMS adopting DREAM model (DREAM DBMS) and constructing function of shapes. However, shape is imperfect to describe database structures because it can't explain nested structures of elements. In this paper, we will profuse a "shape graph"that is dynamic schema showing database structures more exactly and extend the DREAM DBMS. Further we will evaluate the performance of constructing function of shapes and shape graphs.

• Korean Journal of Soil Science and Fertilizer
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• v.42 no.spc
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• pp.51-56
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• 2009

Numerous methods can be employed to teach students and farmers on soil science and fertilizer. The cartoon is one of the most effective teaching aids both for instructors and students. Instructor becomes a director of the cartoon to organize the framework of story based on the chapter of lecture. Director should adapt the regular sentence on soil science into the short and concise scenario suitable for the visual expression of the cartoon, followed by the selection of the scene for drawing. Next is to cast the instructor for the role of actor or sometimes supporter. After casting, the director starts to draw each cut of the cartoon in sequence. I published the cartoons in weekly serials in the farmer's newspaper for three years. Each cartoon was composed of eighteen cuts and formatted with the introductory, main and conclusive issues. Cartoon started with dialogues between the cast on subject to stimulate the interest by farmers, inserted the tables and graphs for the scientific knowledge and summarized the chapter. Major contents covered in the cartoons were glossary, nutrient uptake, pH, salt accumulation, moisture, pollution, chemical fertilizer, compost and other general aspects of soils. Cartoons had been very popular serial of the newspaper and are archived in the web for reviewing.