• Journal of applied mathematics & informatics
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• v.41 no.3
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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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• v.25 no.2
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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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• v.5 no.1_2
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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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• v.42 no.1
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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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• v.21 no.2
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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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• v.24 no.1
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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.

• Korean journal of food and cookery science
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• v.20 no.1
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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％).

• The Korean Journal of Nuclear Medicine
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• v.38 no.4
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• pp.300-305
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• 2004

Purpose: We underwent this study to evaluate the factors which influence labeling efficiency when modified in vivo erythrocyte labeling technique was used. Materials and methods: Thirty healthy volunteers (M:F=19:11, age:$25{\pm}2$ yrs) were enrolled in this study. Totally, two hundred ten samples were obtained from them. The 1 mg of stannous pyrophosphate was injected intravenously at the beginning of labeling. After suitable tinning time (5 min, 20 min, 35 min) passed by, blood (5 mL, 3 mL or 1 mL) was withdrawn into 10 mL syringe previously containing Tc-99m (740 MBq) and anticoagulant (heparin, ACD or CPDA) through 19-gauged scalp needle. The generator ingrowth time of Tc-99m was within 24 hrs in each case. The blood samples were placed on rotating invertor during incubation (10 min, 25 min, 40 min) but some of them were not. Immediately after the conclusion of incubation, the labeled blood specimens to analyze were centrifuged. and then %Unbound Tc-99m was calculated. Statical analysis was used paired T-test and one way ANOVA with SPSS 10.0. Results: The binding efficiency at 1 mL of blood volume was $73{\pm}32%,\;91{\pm}10%$ at 3 mL and $96{\pm}7%$ at 5 mL (p<0.01). The binding efficiency at 5 min of tinning time was $45{\pm}23%,\;98{\pm}6%$, at 20 min and $97{\pm}8%$ at 35 min (p<0.001). The binding efficiency at 10 min of incubation time was $96{\pm}7%,\;95{\pm}12%$ at 25 min and $98{\pm}3%$ at 40 min (p>0.05). The binding efficiency in case of using rotating invertor was $96{\pm}7%$ and the binding efficiency in case of not using it was $87{\pm}18%$ (p>0.05). There was no significant difference between them. In binding efficiency according to kinds of anticoagulants, ACD was $98{\pm}4%$, CPDA was $97{\pm}6%$ and heparin was $89{\pm}20%$ (p<0.001). Conclusion: When modified in vivo erythrocyte labeling technique is used with Tc-99m, the methods to obtain the highest labeling efficiency are as follow. The withdrawing blood volume should be over 3 mL, tinning time should be kept between 20 min and 35 min, and incubation time should be kept between 10 min and 40 min. ACD or CPDA have to be used as a anticoagulant except heparin and the blood samples should be placed on rotating invertor during incubation.

• Nuclear Medicine and Molecular Imaging
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• v.43 no.4
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• pp.330-336
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• 2009

Purpose: We established radiolabeling conditions of NOTA and DOTA with a generator-produced PET radionuclide $^{68}$Ga and studied in vitro characteristics such as stability, serum protein binding, octanol/water distribution, and interference with other metal ions. Materials and Methods: Various concentrations of NOTA 3HCl and DOTA 4HCl were labeled with 1 mL $^{68}$GaCl$_3$ (0.18$\sim$5.75 mCi in 0.1 M HCl in various pH. NOTA 3HCl (0.373 mM) was labeled with $^{68}$GaCl$_3$(0.183$\sim$0.232 mCi/0.1 M HCl 1.0 mL) in the presence of CuCl$_2$, FeCl$_2$, InCl$_3$, FeCl$_3$, GaCl$_3$, MgCl$_2$ or CaCl$_2$ (0$\sim$6.07 mM) at room temperature. The labeling efficiencies of $^{68}$Ga-NOTA and $^{68}$Ga-DOTA were checked by ITLC-SG using acetone or saline as mobile phase. Stabilities, protein bindings, and octanol distribution coefficients of the labeled compounds also were investigated. Results: $^{68}$Ga-NOTA and $^{68}$Ga-DOTA were labeled optimally at pH 6.5 and pH 3.5, respectively, and the chelates were stable for 4 hr either in the reaction mixture at room temperature or in the human serum at 37$^{\circ}C$. NOTA was labeled at room temperature while DOTA required heating for labeling. $^{68}$Ga-NOTA labeling efficiency was reduced by CuCl$_2$, FeCl$_2$, InCl$_2$, FeCl$_3$ or CaCl$_3$, however, was not influenced by MgCl$_2$ or CaCl$_2$. The protein binding was low (2.04$\sim$3.32%). Log P value of $^{68}$Ga-NOTA was -3.07 indicating high hydrophilicity. Conclusion: We found that NOTA is a better bifunctional chelating agent than DOTA for $^{68}$Ga labeling. Although, $^{68}$Ga-NOTA labeling is interfered by various metal ions, it shows high stability and low serum protein binding.

• Journal of Radiopharmaceuticals and Molecular Probes
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• v.5 no.1
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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.