• Journal of the Korean Mathematical Society
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• v.44 no.6
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• pp.1351-1362
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• 2007

We enumerate the number of weighted complete graphs and compute its generating function.

• Journal of applied mathematics & informatics
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• v.22 no.1_2
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• pp.435-440
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• 2006

Let G be a simple graph with vertex set V(G) and edge set E(G). A subset S of E(G) is called an edge cover of G if the subgraph induced by S is a spanning subgraph of G. The maximum number of edge covers which form a partition of E(G) is called edge covering chromatic number of G, denoted by X'c(G). It is known that for any graph G with minimum degree ${\delta},\;{\delta}-1{\le}X'c(G){\le}{\delta}$. If $X'c(G) ={\delta}$, then G is called a graph of CI class, otherwise G is called a graph of CII class. It is easy to prove that the problem of deciding whether a given graph is of CI class or CII class is NP-complete. In this paper, we consider the classification of nearly bipartite graph and give some sufficient conditions for a nearly bipartite graph to be of CI class.

• Bulletin of the Korean Mathematical Society
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• v.44 no.4
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• pp.709-719
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• 2007

In [8], it is shown that the complete multipartite graph $K_{n(2t)}$ having n partite sets of size 2t, where $n{\geq}6\;and\;t{\geq}1$, has a decomposition into gregarious 6-cycles if $n{\equiv}0,1,3$ or 4 (mod 6). Here, a cycle is called gregarious if it has at most one vertex from any particular partite set. In this paper, when $n{\equiv}0$ or 3 (mod 6), another method using difference set is presented. Furthermore, when $n{\equiv}0$ (mod 6), the decomposition obtained in this paper is ${\infty}-circular$, in the sense that it is invariant under the mapping which keeps the partite set which is indexed by ${\infty}$ fixed and permutes the remaining partite sets cyclically.

• Communications of the Korean Mathematical Society
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• v.39 no.3
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• pp.803-819
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• 2024

In our study, the integration of fuzzy graphs into classical graph theory gives rise to a novel concept known as "Fuzzy Super Subdivision." Let SS_f (G) be the fuzzy super subdivision graphs, by substituting a complete bipartite graph k_(2,m) (m = 1, 2, . . .) for each edge of a fuzzy graph. The attributes and properties of this newly proposed concept are briefly outlined, in addition to illustrative examples. Furthermore, significant findings are discussed on connectivity, size, degree and order of fuzzy super subdivision structures. To illustrate the practical implications of our approach, we present an application focused on analyzing the growth of infections in blood or urine samples using the Fuzzy Super Subdivision model.

• Korean Journal of Mathematics
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• v.31 no.4
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• pp.485-494
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• 2023

The degree of hesitancy of a vertex in a hesitancy fuzzy graph depends on the degree of membership and non-membership of the vertex. We define a new class of hesitancy fuzzy graph, the intuitionistic hesitancy fuzzy graph in which the degree of hesitancy of a vertex is independent of the degree of its membership and non-membership. We introduce the idea of β-product of a pair of hesitancy fuzzy graphs and intuitionistic hesitancy fuzzy graphs and prove certain results based on this product.

• Bulletin of the Korean Mathematical Society
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• v.59 no.2
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• pp.345-350
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• 2022

In this work, we confirm a weak version of a conjecture proposed by Hong Wang. The ideal of the work comes from the tree packing conjecture made by Gyárfás and Lehel. Bollobás confirms the tree packing conjecture for many small tree, who showed that one can pack T₁, T₂, …, $T_{n/\sqrt{2}}$ into Kn and that a better bound would follow from a famous conjecture of Erdős. In a similar direction, Hobbs, Bourgeois and Kasiraj made the following conjecture: Any sequence of trees T₁, T₂, …, T_n, with T_i having order i, can be packed into K_n-1,[n/2]. Further Hobbs, Bourgeois and Kasiraj [3] proved that any two trees can be packed into a complete bipartite graph K_n-1,[n/2]. Motivated by the result, Hong Wang propose the conjecture: For each k-partite tree T(𝕏) of order n, there is a restrained packing of two copies of T(𝕏) into a complete k-partite graph B_n+m(𝕐), where $m={\lfloor}{\frac{k}{2}}{\rfloor}$. Hong Wong [4] confirmed this conjecture for k = 2. In this paper, we prove a weak version of this conjecture.

• Journal of applied mathematics & informatics
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• v.34 no.3_4
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• pp.341-354
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• 2016

In this paper, the exact formulae for the generalized product degree distance, reciprocal product degree distance and product degree distance of tensor product of a connected graph and the complete multipartite graph with partite sets of sizes m₀, m₁, ⋯ , m_r−1 are obtained.

• Proceedings of the Korean Society for Bioinformatics Conference
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• 2005.09a
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• pp.267-271
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• 2005

The goal of data mining is to extract new and useful knowledge from large scale datasets. As the amount of available data grows explosively, it became vitally important to develop faster data mining algorithms for various types of data. Recently, an interest in developing data mining algorithms that operate on graphs has been increased. Especially, mining frequent patterns from structured data such as graphs has been concerned by many research groups. A graph is a highly adaptable representation scheme that used in many domains including chemistry, bioinformatics and physics. For example, the chemical structure of a given substance can be modelled by an undirected labelled graph in which each node corresponds to an atom and each edge corresponds to a chemical bond between atoms. Internet can also be modelled as a directed graph in which each node corresponds to an web site and each edge corresponds to a hypertext link between web sites. Notably in bioinformatics area, various kinds of newly discovered data such as gene regulation networks or protein interaction networks could be modelled as graphs. There have been a number of attempts to find useful knowledge from these graph structured data. One of the most powerful analysis tool for graph structured data is frequent subgraph analysis. Recurring patterns in graph data can provide incomparable insights into that graph data. However, to find recurring subgraphs is extremely expensive in computational side. At the core of the problem, there are two computationally challenging problems. 1) Subgraph isomorphism and 2) Enumeration of subgraphs. Problems related to the former are subgraph isomorphism problem (Is graph A contains graph B?) and graph isomorphism problem(Are two graphs A and B the same or not?). Even these simplified versions of the subgraph mining problem are known to be NP-complete or Polymorphism-complete and no polynomial time algorithm has been existed so far. The later is also a difficult problem. We should generate all of 2$^n$ subgraphs if there is no constraint where n is the number of vertices of the input graph. In order to find frequent subgraphs from larger graph database, it is essential to give appropriate constraint to the subgraphs to find. Most of the current approaches are focus on the frequencies of a subgraph: the higher the frequency of a graph is, the more attentions should be given to that graph. Recently, several algorithms which use level by level approaches to find frequent subgraphs have been developed. Some of the recently emerging applications suggest that other constraints such as connectivity also could be useful in mining subgraphs : more strongly connected parts of a graph are more informative. If we restrict the set of subgraphs to mine to more strongly connected parts, its computational complexity could be decreased significantly. In this paper, we present an efficient algorithm to mine frequent subgraphs that are more strongly connected. Experimental study shows that the algorithm is scaling to larger graphs which have more than ten thousand vertices.

• Journal of applied mathematics & informatics
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• v.24 no.1_2
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• pp.261-271
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• 2007

The symbol C($m_1^{n_1}m_2^{n_2}{\cdots}m_s^{n_s}$) denotes a 2-regular graph consisting of $n_i$ cycles of length $m_i,\;i=1,\;2,\;{\cdots},\;s$. In this paper, we give some construction methods of cyclic($K_v$, G)-designs, and prove that there exists a cyclic($K_v$, G)-design when $G=C((4m_1)^{n_1}(4m_2)^{n_2}{\cdots}(4m_s)^{n_s}\;and\;v{\equiv}1(mod\;2|G|)$.

• Journal of The Korean Association of Information Education
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• v.2 no.2
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• pp.269-277
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• 1998

In most of the distributed deadlock detection algorithms using messages called probes, only a portion of the generated messages are effectively used, and hence the wasted probes cause heavy communication traffic. In this paper, a distributed deadlock detection algorithm is proposed which can efficiently detect deadlocks making use of those residue probes. Our algorithm is complete in the sense that they detect not only those deadlocks in which the initiator is involved as most other algorithms do, but all the other deadlocks that are present anywhere in a connected wait-for-graph. To detect all the deadlocks, the algorithms known to be most efficient require O(ne) messages, where e and n are the number of edges and nodes in the graph, respectively. The single execution of the presented algorithm can accomplish the same task with O(e) messages.