• Journal of applied mathematics & informatics
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• 제41권3호
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• pp.511-524
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• 2023

An L(p₁, p₂, p₃, . . . , p_m)-labeling of a graph G is an assignment of non-negative integers, called as labels, to the vertices such that the vertices at distance i should have at least pi as their label difference. If p₁ = 4, p₂ = 3, p₃ = 2, p₄ = 1, then it is called a L(4, 3, 2, 1)-labeling which is widely studied in the literature. A L(4, 3, 2, 1)-path coloring of graphs, is a labeling g : V (G) → Z⁺ such that there exists at least one path P between every pair of vertices in which the labeling restricted to this path is a L(4, 3, 2, 1)-labeling. This concept was defined and results for some simple graphs were obtained by the same authors in an earlier article. In this article, we study the concept of L(4, 3, 2, 1)-path coloring for complete bipartite graphs, 2-edge connected split graph, Cartesian product and join of two graphs and prove an existence theorem for the same.

• Korean Journal of Mathematics
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• 제25권2호
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• pp.279-301
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• 2017

An L(3, 2, 1)-labeling for the graph G = (V, E) is an assignment f of a label to each vertices of G such that ${\mid}f(u)-f({\upsilon}){\mid}{\geq}4-k$ when $dist(u,{\upsilon})=k{\leq}3$. The L(3, 2, 1)-labeling number, denoted by ${\lambda}_{3,2,1}(G)$, for G is the smallest number N such that there is an L(3, 2, 1)-labeling for G with span N. In this paper, we compute the L(3, 2, 1)-labeling number ${\lambda}_{3,2,1}(G)$ when G is a cylindrical grid, which is the cartesian product $P_m{\Box}C_n$ of the path and the cycle, when $m{\geq}4$ and $n{\geq}138$. Especially when n is a multiple of 4, or m = 4 and n is a multiple of 6, then we have ${\lambda}_{3,2,1}(G)=11$. Otherwise ${\lambda}_{3,2,1}(G)=12$.

• Journal of Applied and Pure Mathematics
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• 제5권1_2호
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• pp.41-53
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• 2023

Let G = (V, E) be a (p, q) graph. Define $${\rho}=\{{\frac{2}{p}},\;{\text{{\qquad} if p is even}}\\{\frac{2}{p-1}},\;{{\text{if p is odd}}$$ and L = {±1, ±2, ±3, … , ±ρ} called the set of labels. Consider a mapping f : V ⟶ L by assigning different labels in L to the different elements of V when p is even and different labels in L to p-1 elements of V and repeating a label for the remaining one vertex when p is odd.The labeling as defined above is said to be a pair difference cordial labeling if for each edge uv of G there exists a labeling |f(u) - f(v)| such that ${\mid}{\Delta}_{f_1}-{\Delta}_{f^c_1}{\mid}{\leq}1$, where ${\Delta}_{f_1}$ and ${\Delta}_{f^c_1}$ respectively denote the number of edges labeled with 1 and number of edges not labeled with 1. A graph G for which there exists a pair difference cordial labeling is called a pair difference cordial graph. In this paper we investigate pair difference cordial labeling behaviour of Petersen graphs P(n, k) like P(n, 2), P(n, 3), P(n, 4).

• Journal of applied mathematics & informatics
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• 제42권1호
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• pp.1-14
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• 2024

Let G = (V, E) be a (p, q) graph. Define $$\begin{cases}\frac{p}{2},\:if\:p\:is\:even\\\frac{p-1}{2},\:if\:p\:is\:odd\end{cases}$$ and L = {±1, ±2, ±3, ···, ±ρ} called the set of labels. Consider a mapping f : V → L by assigning different labels in L to the different elements of V when p is even and different labels in L to p - 1 elements of V and repeating a label for the remaining one vertex when p is odd.The labeling as defined above is said to be a pair difference cordial labeling if for each edge uv of G there exists a labeling |f(u) - f(v)| such that |Δ_f1 - Δ_f^c1| ≤ 1, where Δ_f1 and Δ_f^c1 respectively denote the number of edges labeled with 1 and number of edges not labeled with 1. A graph G for which there exists a pair difference cordial labeling is called a pair difference cordial graph. In this paper we investigate the pair difference cordial labeling behavior of subdivision of some graphs.

• Korean Journal of Mathematics
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• 제21권2호
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• pp.151-159
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• 2013

An $L(j,k)$-labeling of a graph G is a vertex labeling such that the difference of the labels of any adjacent vertices is at least $j$ and that of any vertices of distance two is at least $k$ for given $j$ and $k$. The minimum span of all L(2, 1)-labelings of G is called the ${\lambda}$-number of G and is denoted by ${\lambda}(G)$. In this paper, we find a lower bound of the ${\lambda}$-number of the Cartesian product $K_m{\Box}C_n$ of the complete graph $K_m$ of order $m$ and the cycle $C_n$ of order $n$. In fact, we show that when $n{\geq}3$, ${\lambda}(K_4{\Box}C_n){\geq}7$ and the equality holds if and only if n is a multiple of 8. Moreover when $m{\geq}5$, ${\lambda}(K_m{\Box}C_n){\geq}2m-1$ and the equality holds if and only if $n$ is even.

• ALGAE
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• 제24권1호
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• pp.39-45
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• 2009

The conjugation processes of a filamentous freshwater green alga Spirogyra varians were examined using FITC-lectins. Conjugation comprised five steps: 1) aligning with adjacent filaments, 2) formation of conjugation protru-sion (papilla), 3) fusion of the protrusions, 4) formation of conjugation tube,and 5) formation of zygotes. Three lectins, ConA, RCA and UEA, showed considerable labeling during the progression of conjuation. FITC-ConA labeled the surfaces of filaments throughout the whole conjugation processes. FITC-RCA labeling was observed at the conjugation protrusions only after the papilla formation. Strong labeling continued until formationg of zygotes at the contacting area where the conjugation tube developed, but no labeling was detected on the surface of vegetative filaments. The labeling decreased gradually over time and disappeared when zygotes were formed. FITC-UEA showed similar labeling pattern with FITC-RCA except that weak labeling remained after zygote formation. Inhibition experiments using RCA, UEA which are complementary to sugars L-fucose and D-galactose, showed considerable decrease of conjugation (<32% vs. 70% in control). These results suggested that the lectin-carbohydrate recognition system might be involved in the conjugation of spirogyra varians.

• 한국식품조리과학회지
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• 제20권1호
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• pp.100-111
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• 2004

The purpose of this study was to investigate the status of current nutrition labeling on the packaging of the processed foods that provide consumers with a reliable and consistent source of information, which has been considered as a useful aid for food selection and a potent educational tool for nutrition in daily life. The 2,160 processed foods purchased at the supermarket on September, 2002, were divided by food category issued from the 2002 food codes and assessed in the terms of the nutrition composition labeling and nutrition claims. Nutrition composition labeling was found on 356 of the 2160 processed foods items. Milk and dairy products had 49.7％ of nutrition composition labeling, which was the largest number among the food category. Tables were most frequently used as the type of nutrition composition labeling (79.8％). Nutrition composition including many different ways of expression, such as a table of nutrition composition, indication of nutrition composition, analysis table of nutrition composition and comparative table of nutrition composition, made frequent use of nutrition composition labeling titles (78.7％). The various unit of measures were use in the nutrition labeling of the processed foods, per l00g or 100$m\ell$ was the highest (44.6％) under the currently practiced nutrition labeling. The correct labeling standard with nutrient content and ％ RDA except energy, was used on 47.8％ of labels, and those with only liability indication nutrient and liability indication nutrients plus discretion indication nutrients were 25.3 and 22.5％ respectively. The processed foods with nutrition claims were 8.0％ (172 items). Nutrition claims were divided in two ways: nutrient content claims and nutrient comparative claims. The most frequently used claims were contained in the former (44.4％) and more or plus in the latter case (16.3％). Ca was the most popular item as a nutrition claim nutrient (50.6％).

• 대한핵의학회지
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• 제38권4호
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• pp.300-305
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• 2004

목적: 변형 체내 표지법으로 적혈구 표지시 영향을 미칠 수 있는 인자에는 수혈, 환자에게 투여된 약제, 염화주석의 양, 발생기 내부성장 시간, tinning 시간, 배양시간, 혈액 양, 항응고제의 종류, 채취한 혈액을 배양시 rotating invertor 사용 유무 등 매우 다양하다. 저자들은 tinning 시간, 혈액 양, 채취한 혈액을 배양시 rotating invertor 사용 유무, 배양시간, 항응고제의 종류에 따라 변형 체내 표지법으로 적혈구 표지시 높은 결합효율을 유지시키기 위한 최상의 조건을 알아보고자 하였다. 대상 및 방법: 2003년 3월부터 2004년 2월까지 과거에 빈혈, 혈액응고 질환, 출혈성 질환 등의 질병이 없는 자발적인 지원자 30명(남:여=19:11, 연령: $25{\pm}2$세)을 대상으로 하였다. Tinning시간, 혈액 양, 배양시간과 항응고제의 종류에 대한 실험에서는 각각의 인자마다 15명을 대상으로 45개씩, 모두 180개의 혈액 샘플을 얻었고, rotating invertor 사용유무에 대한 실험에서는 15명을 대상으로 30개를 얻어 총 210개의 혈액 샘플을 얻었다. 채취 한 혈액샘플은 원심분리기에서 상층액과 하층액으로 분리시켜 각각의 결합효율을 계산한 후 SPSS 10.0 프로그램을 이용하여 paired T-test와 one way ANOVA통계 검정을 실시하였다. 결과: Tinning 시간에 따른 결합효율은 5분일 때 $45{\pm}23%$, 20분일 때 $97{\pm}8%$, 35분일 때 $98{\pm}6%$로 20분과 35분에서 유의하게 높은 결합효율을 보였다(p<0.001). 혈액 양에 따른 결합효율은 1 mL일 때 $73{\pm}32%$, 3 mL일 때 $91{\pm}10%$, 5 mL일 때 $96{\pm}7%$로 3 mL와 5 mL에서 유의하게 높은 결합효율을 보였다(p<0.05). Rotating invertor를 사용한 경우 결합효율은 $96{\pm}7%$로 사용하지 않은 경우의 $87{\pm}18%$에 비하여 높았으나 통계학적으로 유의한 차이는 없었다(p>0.05). 배양시간에 따른 결합효율은 10분일 때 $96{\pm}7%$, 25분일 때 $95{\pm}12%$, 40분일 때 $98{\pm}3%$로 통계학적으로 유의한 차이는 없었다(p>0.05). 항응고제 종류에 따른 결합효율은 헤파린을 사용한 경우 $89{\pm}20%$, CPDA를 사용한 경우 $97{\pm}6%$, ACD를 사용한 경우 $98{\pm}4%$로 CPDA와ACD를 사용한 경우에 유의하게 높은 결합효율을 보였다(p<0.001). 결론: 변형 체내 표지법으로 적혈구를 표지시 우수한 결합효율을 유지하기 위해서는 채취하는 혈액의 양은 3 mL 이상, 배양시간은 10분 이상(10분-40분), 항응고제는 ACD나 CPDA tinning 시간은 20분 이상(20-35분)을 유지하고, 가능한 rotating invertor를 사용하는 것이 좋을 것으로 생각된다.

• Nuclear Medicine and Molecular Imaging
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• 제43권4호
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• pp.330-336
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• 2009

목적: $^{68}$Ge/$^{68}$Ga-제너레이터에서 생산되는 PET용 방사성동위원소인 $^{68}$Ga을 NOTA와 DOTA에 표지하는 조건을 확립하고 이의 안정성 및 단백질 결합 특성을 연구하였고, 여러 가지의 금속이온 공존시 표지효율에 미치는 영향도 관찰하였다. 대상 및 방법: 여러 가지 농도의 NOTA 3HCl과 DOTA 4HCl에 $^{68}$Ge/$^{68}$Ga-제너레이터에서 0.1 M HCl로 용출한 $^{68}$GaCl$_3$ 6.66$\sim$272.8 MBq 1.0 mL와 합치고 초산나트륨 또는 탄산나트륨 완충액을 사용하여 다양한 pH 조건에서 반응하였다. 다양한 금속이온(CuCl$_2$, FeCl$_2$, InCl$_3$, FeCl$_3$), GaCl$_3$, MgCl$_2$, CaCl$_2$)과 0.373 mM NOTA를 $^{68}$Ga(6.77$\sim$8.58 MBq)으로 표지할 때 표지효율을 관찰하였다. $^{68}$Ga의 표지효율은 ITLC-SG고정상으로 하고 아세톤과 생리식염수를 이동상으로 하여 측정하였다. 최적의 pH 조건에서 $^{68}$Ga-NOTA와 $^{68}$Ga-DOTA를 표지한 후 4 시간 동안 안정성을 확인하고, 사람 혈청에서의 단백질 결합능을 평가하였으며, 지용성 정도를 측정하기 위하여 octhanol distribulion 실험을 실시하여 log P값을 구하였다. 결과: $^{68}$Ga-NOTA와 $^{68}$Ga-DOTA의 치적 표지 pH 조건은 각각 pH 6.5와 3.5였고, NOTA는 실온에서 표지가 잘 되었으나 DOTA는 가열이 필요하였다. MgCl$_2$와 CaCl$_2$의 존재는 $^{68}$Ga-NOTA의 표지 효율에 영향을 미치지 않았으나 CuCl$_2$, FeCl$_2$, InCl$_3$, FeCl$_3$, GaCl$_3$이 존재할 경우에는 표지효율이 감소하였다. $^{68}$Ga-NOTA와 $^{68}$Ga-DOTA는 실온에 그대로 두거나 사람혈청과 37$^{\circ}C$에 두었을 때 4 시간 이상 안정하였고, 사람 혈청 단백질 결합능은 2.04$\sim$3.32%로 낮았으며, log P 값은 -3.07로 수용성을 보였다. 결론: $^{68}$Ga의 표지에는 NOTA가 DOTA에 비하여 이상적인 양기능성 킬레이트제로 쓰일 수 있음을 알았다. 또한 $^{68}$Ga-NOTA는 금속이온 존재시 표지효율이 떨어질 수 있지만 안정하고 낮은 단백질 결합을 보였다.

• 대한방사성의약품학회지
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• 제5권1호
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• pp.18-25
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• 2019

2-Nitroimidazole derivatives have been reported to accumulate in hypoxic tissue. We prepared a novel $^{99m}Tc-MAG_3$-2-nitroimidazole and evaluated the feasibility for hypoxia imaging agent. $Bz-MAG_3$-2-nitroimidazole was synthesized by direct coupling of $Bz-MAG_3$ and 2-nitroimidazole using dicyclohexylcarbodiimide. $Bz-MAG_3$-2-nitroimidazole was labeled with $^{99m}Tc$ in the presence of tartaric acid and $SnCl_2-2H_2O$ at $100^{\circ}C$ for 30 min. And the reaction mixture was purified by $C_{18}$ Sep-pak cartridge. The labeling efficiency and the radiochemical purity were checked by ITLC-SG/acetonitrile. The tumor was grown in balb/c mice for 8~13 days after the subcutaneous injection of tumor cells, CT-26 (murine colon adenocarcinoma cell). Biodistribution study and tumor autoradiography were performed in the xenografted mice after i.v injection of 74 kBq/0.1 mL and 19 MBq/0.1 mL of $^{99m}Tc-MAG_3$-2-nitroimidazole, respectively. In vivo images of $^{99m}Tc-MAG_3$-2-nitroimidazole in tumor bearing mice were obtained 1.5 hr post injection. The labeling efficiency was $45{\pm}20%$ and the radiochemical purity after purification was over 95%. Paper electrophoresis confirmed negative charge of $^{99m}Tc-MAG_3$-2-nitroimidazole. $^{99m}Tc-MAG_3$-2-nitroimidazole was very stable at room temperature and its protein binding was 53%. The $^{99m}Tc-MAG_3$-2-nitroimidazole exhibited high uptake in the liver, stomach and intestine. In biodistribution study using tumor bearing mice, the uptakes (% ID/g) of the tumor were $0.5{\pm}0.1$, $0.4{\pm}0.0$, $0.2{\pm}0.1$ and $0.1{\pm}0.1$ at 5, 15, 30 min and 4 hrs. Tumor/muscle ratio were $1.4{\pm}0.1$, $2.2{\pm}0.83$, $3.0{\pm}0.9$, and 3.7 (n=2) for 5, 15, 30 min and 4 hrs. The uptake in hypoxic area was found higher than in non-hypoxic area of tumor tissue by autoradiography. In vivo images showed the relatively faint uptake to the hypoxic tumor region. $^{99m}Tc-MAG_3$-2-nitroimidazole was successfully synthesized and found feasible for imaging hypoxia.