• Communications of the Korean Mathematical Society
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• v.38 no.3
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• pp.733-740
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• 2023

Let R be a commutative ring and M be an R-module. The dimension graph of M, denoted by DG(M), is a simple undirected graph whose vertex set is Z(M) ⧵ Ann(M) and two distinct vertices x and y are adjacent if and only if dim M/(x, y)M = min{dim M/xM, dim M/yM}. It is shown that DG(M) is a disconnected graph if and only if (i) Ass(M) = {𝖕, 𝖖}, Z(M) = 𝖕 ∪ 𝖖 and Ann(M) = 𝖕 ∩ 𝖖. (ii) dim M = dim R/𝖕 = dim R/𝖖. (iii) dim M/xM = dim M for all x ∈ Z(M) ⧵ Ann(M). Furthermore, it is shown that diam(DG(M)) ≤ 2 and gr(DG(M)) = 3, whenever M is Noetherian with |Z(M) ⧵ Ann(M)| ≥ 3 and DG(M) is a connected graph.

• Journal of the Korean Operations Research and Management Science Society
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• v.19 no.1
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• pp.189-199
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• 1994

This paper considers the covering network design problem (CNDP). In the CNDP, an undirected graph is given where nodes correspond to potential facility sites and arcs to potential links connecting facilities. The objective of the CNDP is to identify the least cost connected subgraph whose nodes cover the given demand points. The problem difines a demand point to be covered if some node in the selected graph is present within an appropriate distance from the demand point. We present an integer programming formulation for the problem and develop a dual-based solution procedure. The computational results for randomly generated test problems are also shown.

• Communications of the Korean Mathematical Society
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• v.18 no.4
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• pp.745-754
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• 2003

Maximum clique problem is to find a maximum clique(largest in size) in an undirected graph G. We present a method that computes either a maximum clique or an upper bound for the size of a maximum clique in G. We show that this method performs well on certain class of graphs and discuss the application of this method in a branch and bound algorithm for solving maximum clique problem, whose efficiency is depended on the computation of good upper bounds.

• The Journal of the Institute of Internet, Broadcasting and Communication
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• v.13 no.1
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• pp.143-152
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• 2013

Given weighted graph G=(V,E), n=|V|, m=|E|, the minimum cut problem is classified with source s and sink t or without s and t. Given undirected weighted graph without s and t, Stoer-Wagner algorithm is most popular. This algorithm fixes arbitrary vertex, and arranges maximum adjacency (MA)-ordering. In the last, the sum of weights of the incident edges for last ordered vertex is computed by cut value, and the last 2 vertices are merged. Therefore, this algorithm runs $\frac{n(n-1)}{2}$ times. Given graph with s and t, Ford-Fulkerson algorithm determines the bottleneck edges in the arbitrary augmenting path from s to t. If the augmenting path is no more exist, we determine the minimum cut value by combine the all of the bottleneck edges. This paper suggests minimum cut algorithm for undirected weighted graph with s and t. This algorithm suggests MA-merging and computes cut value simultaneously. This algorithm runs n-1 times and successfully divides V into disjoint S and V sets on the basis of minimum cut, but the Stoer-Wagner is fails sometimes. The proposed algorithm runs more than Ford-Fulkerson algorithm, but finds the minimum cut value within n-1 processing times.

• Journal of applied mathematics & informatics
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• v.32 no.1_2
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• pp.1-6
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• 2014

Let G = (V ; E) be a simple undirected graph without loops and multiple edges, the vertex and edge sets of it are represented by V = V (G) and E = E(G), respectively. A weighting w of the edges of a graph G induces a coloring of the vertices of G where the color of vertex v, denoted $S_v:={\Sigma}_{e{\ni}v}\;w(e)$. A k-edge-weighting of a graph G is an assignment of an integer weight, w(e) ${\in}${1,2,...,k} to each edge e, such that two vertex-color $S_v$, $S_u$ be distinct for every edge uv. In this paper we determine an exact 3-edge-weighting of complete graphs $k_{3q+1}\;{\forall}_q\;{\in}\;{\mathbb{N}}$. Several open questions are also included.

• Journal of the Korean Mathematical Society
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• v.51 no.1
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• pp.189-202
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• 2014

An element r of a commutative semiring R with identity is said to be identity-summand if there exists $1{\neq}a{\in}R$ such that r+a = 1. In this paper, we introduce and investigate the identity-summand graph of R, denoted by ${\Gamma}(R)$. It is the (undirected) graph whose vertices are the non-identity identity-summands of R with two distinct vertices joint by an edge when the sum of the vertices is 1. The basic properties and possible structures of the graph ${\Gamma}(R)$ are studied.

• Communications for Statistical Applications and Methods
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• v.27 no.5
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• pp.569-578
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• 2020

Various graph clustering methods have been introduced to identify communities in social or biological networks. This paper studies the entropy-based and the Markov chain-based methods in clustering the undirected graph. We examine the performance of two clustering methods with conventional methods based on quality measures of clustering. For the real applications, we collect the mathematical subject classification (MSC) codes of research papers from published mathematical databases and construct the weighted code-to-document matrix for applying graph clustering methods. We pursue to group MSC codes into the same cluster if the corresponding MSC codes appear in many papers simultaneously. We compare the MSC clustering results based on the several assessment measures and conclude that the Markov chain-based method is suitable for clustering the MSC codes.

• Journal of the Korean Operations Research and Management Science Society
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• v.21 no.3
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• pp.215-225
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• 1996

For an undirected stochastic network G, the 2-terminal reliability of G, R(G) is the probability that the specific two nodes (called as terminal nodes) are connected in G. A. typical network reliability problem is to compute R(G). It has been shown that the computation problem of R(G) is NP-hard. So, any algorithm to compute R(G) has a runngin time which is exponential in the size of G. If by some means, the problem size, G is reduced, it can result in immense savings. The means to reduce the size of the problem are the reliability preserving reductions and graph decompositions. We introduce a net set of reliability preserving reductions : the $K^{4}$ (complete graph of 4-nodes)-chain reductions. The total number of the different $K^{4}$ types in R(G), is 6. We present the reduction formula for each $K^{4}$ type. But in computing R(G), it is possible that homeomorphic graphs from $K^{4}$ occur. We devide the homemorphic graphs from $K^{4}$ into 3 types. We develop the reliability preserving reductions for s types, and show that the remaining one is divided into two subgraphs which can be reduced by $K^{4}$-chain reductions 7 polygon-to-chain reductions.

• The Journal of the Institute of Internet, Broadcasting and Communication
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• v.24 no.1
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• pp.149-154
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• 2024

This paper proposes an algorithm that remedy Floyd's the tortoise and the hare algorithm (THA) shortcomings which is specialized in singly linked list (SLL), so this algorithm fails to detect the cycle in undirected graph, digraph, and tree with multiple inputs or outputs. The proposed algorithm simply pruning the source and sink with only one edge using cycle detection of single edge node pruning. As a result of the experimental of various list, undirected graph, digraph, and tree, the proposed algorithm can be successively detect the cycle all of them. Thus, the proposed algorithm has the simplest and fastest advantage in the field of cycle detection.

• Journal of the Korean Institute of Telematics and Electronics
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• v.19 no.6
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• pp.62-66
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• 1982

A method finding out routability for unrouted signal lines and rerouting those which are turned out to be able to route in layout design of LSI is described. In this paper the problems of finding two-disjoint Paths represented by an undirected graph G=(V,E), where V,E are sets of vertices and edges respectively, are studied. The existence of two-disjoint paths from s1, to t1, (called P1) and from S2 to T2 (called P2) indicated by the four vertices on the graph s1, t1, s2, t2 $\in$ V means that two distinct signal lines exist in layout design. It turns out that the proposed time complexity in the algorithm is O (IVI x IEI).