• Journal of the Korean Mathematical Society
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• v.60 no.5
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• pp.959-986
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• 2023

Zhai and Lin recently proved that if G is an n-vertex connected 𝜃(1, 2, r + 1)-free graph, then for odd r and n ⩾ 10r, or for even r and n ⩾ 7r, one has ${\rho}(G){\leq}{\sqrt{{\lfloor}{\frac{n^2}{4}}{\rfloor}}}$, and equality holds if and only if G is $K_{{\lceil}{\frac{n}{2}}{\rceil},{\lfloor}{\frac{n}{2}}{\rfloor}}$. In this paper, for large enough n, we prove a sharp upper bound for the spectral radius in an n-vertex H-free non-bipartite graph, where H is 𝜃(1, 2, 3) or 𝜃(1, 2, 4), and we characterize all the extremal graphs. Furthermore, for n ⩾ 137, we determine the maximum number of edges in an n-vertex 𝜃(1, 2, 4)-free non-bipartite graph and characterize the unique extremal graph.

• 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.

• Journal of applied mathematics & informatics
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• v.42 no.1
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• pp.77-83
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• 2024

Chung defined a pebbling move on a graph G as the removal of two pebbles from one vertex and the addition of one pebble to an adjacent vertex. The monophonic pebbling number guarantees that a pebble can be shifted in the chordless and the longest path possible if there are any hurdles in the process of the supply chain. For a connected graph G a monophonic path between any two vertices x and y contains no chords. The monophonic pebbling number, µ(G), is the least positive integer n such that for any distribution of µ(G) pebbles it is possible to move on G allowing one pebble to be carried to any specified but arbitrary vertex using monophonic a path by a sequence of pebbling operations. The aim of this study is to find out the monophonic pebbling numbers of the sun graphs, (C_n × P₂) + K₁ graph, the spherical graph, the anti-prism graphs, and an n-crossed prism graph.

• The Journal of the Institute of Internet, Broadcasting and Communication
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• v.15 no.1
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• pp.269-276
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• 2015

This paper proves the Erd$\ddot{o}$s-Faber-Lov$\acute{a}$sz conjecture of the vertex coloring problem, which is so far unresolved. The Erd$\ddot{o}$s-Faber-Lov$\acute{a}$sz conjecture states that "the union of k copies of k-cliques intersecting in at most one vertex pairwise is k-chromatic." i.e., x(G)=k. In a bid to prove this conjecture, this paper employs a method in which it determines the number of intersecting vertices and that of cliques that intersect at one vertex so as to count a vertex of the minimum degree ${\delta}(G)$ in the Minimum Independent Set (MIS) if both the numbers are even and to count a vertex of the maximum degree ${\Delta}(G)$ in otherwise. As a result of this algorithm, the number of MIS obtained is x(G)=k. When applied to $K_k$-clique sum intersecting graphs wherein $3{\leq}k{\leq}8$, the proposed method has proved to be successful in obtaining x(G)=k in all of them. To conclude, the Erd$\ddot{o}$s-Faber-Lov$\acute{a}$sz conjecture implying that "the k-number of $K_k$-clique sum intersecting graph is k-chromatic" is proven.

• The Journal of Korean Academy of Prosthodontics
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• v.39 no.1
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• pp.58-70
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• 2001

For the successful treatment of complete denture, obtaining a good retention is essential. There are lots of factors affecting denture retention. Denture material, one of those factors affecting denture retention, was the subject of this study, and internal surface treatment also considered for the method of enhancing denture retention. Two resin(Lucitone $199^{(R)}$(heat cured resin) Vertex $CP^{(R)}$(self cured resin)) and two metal($Biosil^{(R)}$(Co-Cr alloy), $Vitallium^{(R)}$(Co-Cr alloy)) denture base materials were used for making test denture base. Newly developed device was used for measuring denture retention. After the retention was measured. We treated internal surface of test denture base with $50{\mu}m\;Al_2O_3$ powder, under 90psi, for 1 minute. Then the retention was measured again. The result was analyzed with K-S test, one-way ANOVA test and independent t-test to deter mine the significant differences as the 95% level of confidence. The results are as follows : In cases of without internal surface treatment, the retention was increased in order of $Vitallium^{(R)},\;Biosil^{(R)},\;Vertex CP^{(R)}$ and Lucitone $199^{(R)}$. Except between Vertex $CP^{(R)}$ and $Biosil^{(R)}$, retention of the other materials was significantly different (p<0.05). After the treatment of internal surface, the retention was increased in order of $Vitallium^{(R)},\;Biosil^{(R)},\;Lucno\;199^{(R)},\;Vertex\;CP^{(R)}$. Except between Lucitone $199^{(R)}$ and Vertex $CP^{(R)}$, $Vitallium^{(R)}$ and $Biosil^{(R)}$ the retention of remaining groups was significantly different each other (p<0.05). In the matter of each material, after the internal surface treatment the retention was increased with Vertex $CP^{(R)},\;Biosil^{(R)}\;and\;Vitallium^{(R)}$ and the value of differences were statistically significant. When we compare the retention of resin and metal denture base, the retention of both denture bases increased significantly with internal surface treatment, and resin denture base showed better retention. As the results show, selecting denture base material could be an important choice of complete denture treatment. To increase denture retention, internal surface treatment can be considered as a possible method.

• Journal of the Korea Society of Computer and Information
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• v.16 no.7
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• pp.85-93
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• 2011

The Vertex Coloring Problem hasn't been solved in polynomial time, so this problem has been known as NP-complete. This paper suggests linear time algorithm for Vertex Coloring Problem (VCP). The proposed algorithm is based on assumption that we can't know a priori the minimum chromatic number ${\chi}(G)$=k for graph G=(V,E) This algorithm divides Vertices V of graph into two parts as independent sets $\overline{C}$ and cover set C, then assigns the color to $\overline{C}$. The element of independent sets $\overline{C}$ is a vertex ${\upsilon}$ that has minimum degree ${\delta}(G)$ and the elements of cover set C are the vertices ${\upsilon}$ that is adjacent to ${\upsilon}$. The reduced graph is divided into independent sets $\overline{C}$ and cover set C again until no edge is in a cover set C. As a result of experiments, this algorithm finds the ${\chi}(G)$=k perfectly for 26 Graphs that shows the number of selecting ${\upsilon}$ is less than the number of vertices n.

• Journal of applied mathematics & informatics
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• v.17 no.1_2_3
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• pp.643-651
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• 2005

In the first part of this paper we investigate several statements concerning infinite maximal planar graphs which are equivalent in finite case. In the second one, for a given induced $\theta$-path (a finite induced path whose endvertices are adjacent to a vertex of infinite degree) in a 4-connected VAP-free maximal planar graph containing a vertex of infinite degree, a new $\theta$-path is constructed such that the resulting fan is tight.

• Journal of applied mathematics & informatics
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• v.31 no.1_2
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• pp.99-110
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• 2013

In this article, sensitivity analysis of atmospheric dispersion model RIMPUFF is considered. Uncertain parameters are taken to be triangular fuzzy numbers, and sensitivity analysis is carried out by using the Hartley-like measure. Codes for evaluating membership function using the Vertex method and the Hartley-like measure are prepared using Matlab.

• Journal of the Chungcheong Mathematical Society
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• v.19 no.4
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• pp.403-408
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• 2006

A pebbling move on a connected graph G is taking two pebbles off of one vertex and placing one of them on an adjacent vertex. The domination cover pebbling number ${\psi}(G)$ is the minimum number of pebbles required so that any initial configuration of pebbles can be transformed by a sequence of pebbling moves so that the set of vertices that contain pebbles forms a domination set of G. We determine the domination cover pebbling number for fans, fuses, and pseudo-star.

• Proceedings of the IEEK Conference
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• 2000.11d
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• pp.97-100
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• 2000

This paper presents a new approach to the vertex based shape coding technique. The conventional approaches encode objects using a spline method with the same distortion coefficients. The proposed approach, however, classifies the objects based on the object's features, and then applies different distortion values depending on the classified object types. Using this pre-classifying technique, this paper reduces the bit rate and the computational complexity necessary for the encoding process. The performance of the proposed method has been proved by experiments on the various sample Images.