• Journal of the Korean Mathematical Society
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• v.45 no.5
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• pp.1341-1360
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• 2008

In this paper we investigate the projections of bouquet graph B with two cycles. A projection of B is said to be trivial if only trivial embeddings are obtained from the projection. It is shown that, to cover all nontrivial projections of B, at least three embeddings of B are needed. We also show that a nontrivial projection of B is covered by one of some two embeddings if the image of each cycle has at most one self-crossing.

• Communications of the Korean Mathematical Society
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• v.11 no.4
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• pp.1095-1104
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• 1996

We enumerate the congruence classes of 2-cell embeddings of a dartboard graph into surfaces with respect to a group consisting of graph automorphisms of a dartboard graph.

• Journal of the Korean Mathematical Society
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• v.44 no.3
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• pp.661-671
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• 2007

We investigate the number of knots and links in linear embeddings of $K_6$, the complete graph with 6 vertices. Concretely, we show that any linear embedding of $K_6$ contains either only one Hopf link, or three Hopf links and one trefoil knot.

• Honam Mathematical Journal
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• v.36 no.2
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• pp.399-415
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• 2014

We study the Seifert surfaces of a link by relating the embeddings of graphs with induced graphs. As applications, we prove that every link L is the boundary of an oriented surface which is obtained from a graph embedding of a complete bipartite graph $K_{2,n}$, where all voltage assignments on the edges of $K_{2,n}$ are 0. We also provide an algorithm to construct such a graph diagram of a given link and demonstrate the algorithm by dealing with the links $4^2_1$ and $5_2$.

• KSII Transactions on Internet and Information Systems (TIIS)
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• v.14 no.5
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• pp.1909-1928
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• 2020

Predicting human mobility has always been an important task in Location-based Social Network. Previous efforts fail to capture spatial dependence effectively, mainly reflected in weakening the location topology information. In this paper, we propose a neural network-based method which can capture spatial-temporal dependence to predict the next location of a person. Specifically, we involve a graph convolutional network (GCN) based on a seq2seq framework to capture the location topology information and temporal dependence, respectively. The encoder of the seq2seq framework first generates the hidden state and cell state of the historical trajectories. The GCN is then used to generate graph embeddings of the location topology graph. Finally, we predict future trajectories by aggregated temporal dependence and graph embeddings in the decoder. For evaluation, we leverage two real-world datasets, Foursquare and Gowalla. The experimental results demonstrate that our model has a better performance than the compared models.

• Bulletin of the Korean Mathematical Society
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• v.39 no.1
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• pp.175-184
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• 2002

A Cellular embedding of a graph G into an orientable surface S can be considered as a cellular decomposition of S into 0-cells, 1-cells and 2-cells and vise versa, in which 0-cells and 1-cells form a graph G and this decomposition of S is called a map in S with underlying graph G. For a map M with underlying graph G, we define a natural rotation on the line graph of the graph G and we introduce the line map for M. we find that genus of the supporting surface of the line map for a map and we give a characterization for the line map to be embedded in the sphere. Moreover we show that the line map for any life of a map M is map-isomorphic to a lift of the line map for M.

• Korean Journal of Mathematics
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• v.20 no.4
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• pp.403-414
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• 2012

A voltage assignment on a graph was used to enumerate all possible 2-cell embeddings of a graph onto surfaces. The boundary of the surface which is obtained from 0 voltage on every edges of a very special diagram of a complete bipartite graph $K_{m,n}$ is surprisingly the ($m,n$) torus link. In the present article, we prove that every link is the boundary of a complete bipartite multi-graph $K_{m,n}$ for which voltage assignments are either -1 or 1 and that every link is the boundary of a complete bipartite graph $K_{2,n}$ for which voltage assignments are either -1, 0 or 1 where edges in the diagram of graphs may be linked but not knotted.

• Journal of the Korean Mathematical Society
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• v.48 no.2
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• pp.367-382
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• 2011

We establish a necessary and sufficient condition for a heptagonal knot to be figure-8 knot. The condition is described by a set of Radon partitions formed by vertices of the heptagon. In addition we relate this result to the number of nontrivial heptagonal knots in linear embeddings of the complete graph $K_7$ into $\mathbb{R}^3$.

• Proceedings of the Korean Society of Computer Information Conference
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• 2024.01a
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• pp.23-26
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• 2024

본 논문에서는 그래프 구조의 데이터에서 각 노드의 신호를 예측하는 연구를 진행하였다. 이를 위해 분석하고자 하는 그래프에 대해 연결 관계를 기반으로 각 노드에 비-유클리드 공간 상에서의 좌표를 부여하여 그래프의 공간적 임베딩을 얻은 뒤, 각 노드의 공간적 임베딩을 입력으로 받고 해당 노드의 신호를 예측하는 그래프 암시적 신경 표현 모델을 제안 하였다. 제안된 모델의 검증을 위해 네트워크형 데이터와 3차원 메시 데이터 두 종류의 그래프 데이터에 대하여 신호 학습, 신호 예측 및 메시 데이터의 초해상도 과정 실험들을 진행하였다. 전반적으로 기존의 그래프 암시적 신경 표현 모델과 비교하였을 때 비슷하거나 더 우수한 성능을 보였으며, 특히 네트워크형 그래프 데이터 신호 예측 실험에서 큰 성능 향상을 보였다.

• Proceedings of the IEEK Conference
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• 2000.07b
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• pp.731-734
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• 2000

We present a novel graph labeling problem called Fibonacci edge labeling. The constraint in this labeling is placed on the allowable edge label which is the difference between the labels of endvertices of an edge. Each edge label should be (3m+2)-th Fibonacci numbers. We show that every Fibonacci tree can be labeled Fibonacci edge labeling. The labelings on the Fibonacci trees are applied to their embeddings into Fibonacci Circulants.