• 대한수학회보
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• 제50권4호
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• pp.1303-1314
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• 2013

An injective coloring of a graph G is an assignment of colors to the vertices of G so that any two vertices with a common neighbor receive distinct colors. A graph G is said to be injectively $k$-choosable if any list $L(v)$ of size at least $k$ for every vertex $v$ allows an injective coloring ${\phi}(v)$ such that ${\phi}(v){\in}L(v)$ for every $v{\in}V(G)$. The least $k$ for which G is injectively $k$-choosable is the injective choosability number of G, denoted by ${\chi}^l_i(G)$. In this paper, we obtain new sufficient conditions to be ${\chi}^l_i(G)={\Delta}(G)$. Maximum average degree, mad(G), is defined by mad(G) = max{2e(H)/n(H) : H is a subgraph of G}. We prove that if mad(G) < $\frac{8k-3}{3k}$, then ${\chi}^l_i(G)={\Delta}(G)$ where $k={\Delta}(G)$ and ${\Delta}(G){\geq}6$. In addition, when ${\Delta}(G)=5$ we prove that ${\chi}^l_i(G)={\Delta}(G)$ if mad(G) < $\frac{17}{7}$, and when ${\Delta}(G)=4$ we prove that ${\chi}^l_i(G)={\Delta}(G)$ if mad(G) < $\frac{7}{3}$. These results generalize some of previous results in [1, 4].

• 한국공간구조학회논문집
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• 제8권4호
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• pp.81-90
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• 2008

이 논문에서는 적분형 르장드르 다항식을 사용한 3차원 계층적 고체요소의 유한요소 정식화를 보여준다. 제안하는 육면체 고체요소는 절점, 변, 면, 그리고 내부모우드를 포함한은 4개의 서로 다른 모우드로 구성되어 있다. 영에너지 모우드와 일정변형률 조건을 확인하기 위해 고유치 시험과 조각시험이 수행되었다. 여기에 추가되어, 적응적 p-유한요소해석을 위해 유한요소해석으로부터 구한 후처리 응력값의 평활화에 기초를 둔 사후오차평가 기법이 연구된다. 자유도가 증가함에 따라 수렴속도측면에서 균등 p-분배와 불균등 p-분배에 의한 유한요소해의 차이점이 비교된다. 제안된 요소의 성능을 보이기 위해 간단한 캔틸레버보가 테스트되었다.

• 대한수학회보
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• 제50권1호
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• pp.117-123
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• 2013

Let G be a finite non-abelian group. We define the non-commuting graph ${\nabla}(G)$ of G as follows: the vertex set of ${\nabla}(G)$ is G-Z(G) and two vertices x and y are adjacent if and only if $xy{\neq}yx$. In this paper we prove that if G is a finite group with $${\nabla}(G){\simeq_-}{\nabla}(\mathbb{A}_{22})$$, then $$G{\simeq_-}\mathbb{A}_{22}$$where $\mathbb{A}_{22}$ is the alternating group of degree 22.

• Journal of applied mathematics & informatics
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• 제31권3_4호
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• pp.533-544
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• 2013

The multiplicative version of Wiener index (${\pi}$-index), proposed by Gutman et al. in 2000, is equal to the product of the distances between all pairs of vertices of a (molecular) graph G. In this paper, we first present some sharp bounds in terms of the order and other graph parameters including the diameter, degree sequence, Zagreb indices, Zagreb coindices, eccentric connectivity index and Merrifield-Simmons index for ${\pi}$-index of general connected graphs and trees, as well as a Nordhaus-Gaddum-type bound for ${\pi}$-index of connected triangle-free graphs. Then we study the behavior of ${\pi}$-index upon the case when removing a vertex or an edge from the underlying graph. Finally, we investigate the extremal properties of ${\pi}$-index within the set of trees and unicyclic graphs.

• Korean Journal of Mathematics
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• 제28권3호
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• pp.405-419
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• 2020

The zeroth-order general Randić index ⁰R_α(G) of the graph G is defined as ∑_u∈V(G)d(u)^α, where d(u) is the degree of vertex u and α is an arbitrary real number. In this paper, the maximum value of zeroth-order general Randić index on the graphs of order n with a given clique number is presented for any α ≠ 0, 1 and α ∉ (2, 2n-1], where n = |V (G)|. The minimum value of zeroth-order general Randić index on the graphs with a given clique number is also obtained for any α ≠ 0, 1. Furthermore, the corresponding extremal graphs are characterized.

• 대한수학회논문집
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• 제33권2호
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• pp.631-638
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• 2018

Chellai et al. [3] gave an upper bound on the [1, 2]-domination number of tree and posed an open question "how to classify trees satisfying the sharp bound?". Yang and Wu [5] gave a partial solution for tree of order n with ${\ell}$-leaves such that every non-leaf vertex has degree at least 4. In this paper, we give a new upper bound on the [1, 2]-domination number of tree which extends the result of Yang and Wu. In addition, we design a polynomial time algorithm for solving the open question. By using this algorithm, we give a characterization on the [1, 2]-domination number for trees of order n with ${\ell}$ leaves satisfying $n-{\ell}$. Thereby, the open question posed by Chellai et al. is solved.

• Korean Journal of Mathematics
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• 제23권3호
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• pp.401-408
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• 2015

The first and second Zagreb indices of a graph G are defined as $M_1(G)={\sum}_{{\nu}{\in}V}d_G({\nu})^2$ and $M_2(G)={\sum}_{u{\nu}{\in}E(G)}d_G(u)d_G({\nu})$. where $d_G({\nu})$ is the degree of the vertex ${\nu}$. G is called a k-apex tree if k is the smallest integer for which there exists a subset X of V (G) such that ${\mid}X{\mid}$ = k and G-X is a tree. In this paper, we determine the maximum Zagreb indices in the class of all k-apex trees of order n and characterize the corresponding extremal graphs.

• 한국통신학회논문지
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• 제26권10B호
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• pp.1399-1408
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• 2001

본 논문에서는 삼각형 메쉬(triangular mesh)로 이루어진 3차원 모델의 연결 정보(connectivity data)와 형상 정보(geometry data)를 삼각형 팬(triangle fan) 구조에 기반하여 효율적으로 압축하는 부호화 기법을 제안한다. 첫째로, 연결 정보의 무손실 부호화로 꼭지점 차수 왜곡(vertex degree warping) 기법을 제안한다. 기존의 알고리듬이 연결 정보와 형상 정보를 분리하여 부호화하는데 반해 제안하는 알고리듬은 연결 정보를 부호화하는데 형상 정보를 이용하여 압축 효율을 높인다. 둘째로, 형상 정보를 압축하기 위해 이중 평행사변형 예측(dual parallelogram prediction) 기법을 제안한다. 삼각형 팬 구조를 이용함으로써 기존의 삼각형 스트립(triangle strip) 기반의 알고리듬보다 정확한 형상 예측이 가능하고, 예측 오차가 작아지게 된다. 다양한 3차원 메쉬 모델에 대한 모의 실험을 통하여 제안하는 알고리듬이 기존의 알고리듬보다 우수한 압축 성능을 나타냄을 확인하다.

• Journal of applied mathematics & informatics
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• 제26권5_6호
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• pp.1085-1100
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• 2008

A set D of vertices of a graph G = (V(G),E(G)) is called a dominating set if every vertex of V(G) - D is adjacent to at least one element of D. The domination number of G, denoted by ${\gamma}(G)$, is the size of its smallest dominating set. Haynes et al.[5] present a conjecture: For any graph G with ${\delta}(G){\geq}k$,$\gamma(G){\leq}\frac{k}{3k-1}n$. When $k\;{\neq}\;6$, the conjecture was proved in [7], [8], [10], [12] and [13] respectively. In this paper we prove that every graph G on n vertices with ${\delta}(G)\;{\geq}\;6$ has a dominating set of order at most $\frac{6}{17}n$. Thus the conjecture was completely proved.

• Journal of applied mathematics & informatics
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• 제29권1_2호
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• pp.451-457
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• 2011

The function f(z) = $e^{p(z)}$ where p(z) is a polynomial of degree n has 2n Julia lines. Julia lines of $e^{p(z)}$ divide the complex plane into 2n equal sectors with the same vertex at the origin. In each sector, $e^{p(z)}$ has radial limits of 0 or innity. Main results of the paper are concerned with maximum curves of $e^{p(z)}$. We deal with some properties of maximum curves of $e^{p(z)}$ and we give some examples of the maximum curves of functions of the form $e^{p(z)}$.