Radio Labeling for Strong Product K3 ⊠ Pn

Abstract. Let denote the graph (k times) where is the strong product of the two graphs G and H. In this paper we prove the conjecture of J. Zaks [3]: For every connected graph G with at least two vertices there exists an integer k = k(G) for which the graph is hamiltonian.

Download Full-text

Homocysteine as a Risk Factor for Atherosclerosis: Is Its Conversion toS-Adenosyl-L-Homocysteine the Key to Deregulated Lipid Metabolism?

Journal of Lipids ◽

10.1155/2011/702853 ◽

2011 ◽

Vol 2011 ◽

pp. 1-11 ◽

Cited By ~ 21

Author(s):

Oksana Tehlivets

Keyword(s):

Risk Factor ◽

Lipid Metabolism ◽

Mammalian Cells ◽

Yeast Cells ◽

Sterol Synthesis ◽

Unfolded Protein ◽

Strong Product ◽

The Unfolded Protein Response ◽

Protein Response

Homocysteine (Hcy) has been recognized for the past five decades as a risk factor for atherosclerosis. However, the role of Hcy in the pathological changes associated with atherosclerosis as well as the pathological mechanisms triggered by Hcy accumulation is poorly understood. Due to the reversal of the physiological direction of the reaction catalyzed byS-adenosyl-L-homocysteine hydrolase Hcy accumulation leads to the synthesis ofS-adenosyl-L-homocysteine (AdoHcy). AdoHcy is a strong product inhibitor ofS-adenosyl-L-methionine (AdoMet)-dependent methyltransferases, and to date more than 50 AdoMet-dependent methyltransferases that methylate a broad spectrum of cellular compounds including nucleic acids, proteins and lipids have been identified. Phospholipid methylation is the major consumer of AdoMet, both in mammals and in yeast. AdoHcy accumulation induced either by Hcy supplementation or due toS-adenosyl-L-homocysteine hydrolase deficiency results in inhibition of phospholipid methylation in yeast. Moreover, yeast cells accumulating AdoHcy also massively accumulate triacylglycerols (TAG). Similarly, Hcy supplementation was shown to lead to increased TAG and sterol synthesis as well as to the induction of the unfolded protein response (UPR) in mammalian cells. In this review a model of deregulation of lipid metabolism in response to accumulation of AdoHcy in Hcy-associated pathology is proposed.

Download Full-text

The metric dimension of strong product graphs

Carpathian Journal of Mathematics ◽

10.37193/cjm.2015.02.15 ◽

2015 ◽

Vol 31 (2) ◽

pp. 261-268

Author(s):

JUAN A. RODRIGUEZ-VELAZQUEZ ◽

◽

DOROTA KUZIAK ◽

ISMAEL G. YERO ◽

JOSE M. SIGARRETA ◽

...

Keyword(s):

Connected Graph ◽

Metric Dimension ◽

Np Hard ◽

Strong Product ◽

Product Graphs ◽

Metric Basis

For an ordered subset S = {s1, s2, . . . sk} of vertices in a connected graph G, the metric representation of a vertex u with respect to the set S is the k-vector r(u|S) = (dG(v, s1), dG(v, s2), . . . , dG(v, sk)), where dG(x, y) represents the distance between the vertices x and y. The set S is a metric generator for G if every two different vertices of G have distinct metric representations with respect to S. A minimum metric generator is called a metric basis for G and its cardinality, dim(G), the metric dimension of G. It is well known that the problem of finding the metric dimension of a graph is NP-Hard. In this paper we obtain closed formulae and tight bounds for the metric dimension of strong product graphs.

Download Full-text

On $ {L}(2,1) $-labelings of some products of oriented cycles

Mathematical Foundations of Computing ◽

10.3934/mfc.2021029 ◽

2021 ◽

Vol 0 (0) ◽

pp. 0

Author(s):

Lucas Colucci ◽

Ervin Győri

Keyword(s):

Cartesian Product ◽

Strong Product

<p style='text-indent:20px;'>We refine two results of Jiang, Shao and Vesel on the <inline-formula><tex-math id="M2">\begin{document}$ L(2,1) $\end{document}</tex-math></inline-formula>-labeling number <inline-formula><tex-math id="M3">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula> of the Cartesian and the strong product of two oriented cycles. For the Cartesian product, we compute the exact value of <inline-formula><tex-math id="M4">\begin{document}$ \lambda(\overrightarrow{C_m} \square \overrightarrow{C_n}) $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M5">\begin{document}$ m $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M6">\begin{document}$ n \geq 40 $\end{document}</tex-math></inline-formula>; in the case of strong product, we either compute the exact value or establish a gap of size one for <inline-formula><tex-math id="M7">\begin{document}$ \lambda(\overrightarrow{C_m} \boxtimes \overrightarrow{C_n}) $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M8">\begin{document}$ m $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M9">\begin{document}$ n \geq 48 $\end{document}</tex-math></inline-formula>.</p>

Download Full-text

On L h , k -Labeling Index of Inverse Graphs Associated with Finite Cyclic Groups

Journal of Mathematics ◽

10.1155/2021/5583433 ◽

2021 ◽

Vol 2021 ◽

pp. 1-7

Author(s):

K. Mageshwaran ◽

G. Kalaimurugan ◽

Bussakorn Hammachukiattikul ◽

Vediyappan Govindan ◽

Ismail Naci Cangul

Keyword(s):

Cyclic Group ◽

Upper Bound ◽

Labeling Index ◽

Positive Difference ◽

Cyclic Groups ◽

Finite Cyclic Group ◽

Minimum Number ◽

The Difference ◽

Radio Labeling

An L h , k -labeling of a graph G = V , E is a function f : V ⟶ 0 , ∞ such that the positive difference between labels of the neighbouring vertices is at least h and the positive difference between the vertices separated by a distance 2 is at least k . The difference between the highest and lowest assigned values is the index of an L h , k -labeling. The minimum number for which the graph admits an L h , k -labeling is called the required possible index of L h , k -labeling of G , and it is denoted by λ k h G . In this paper, we obtain an upper bound for the index of the L h , k -labeling for an inverse graph associated with a finite cyclic group, and we also establish the fact that the upper bound is sharp. Finally, we investigate a relation between L h , k -labeling with radio labeling of an inverse graph associated with a finite cyclic group.

Download Full-text

Computation of Differential in Strong Product Network of Paths via Recursive Iteration

Journal of Physics Conference Series ◽

10.1088/1742-6596/2025/1/012050 ◽

2021 ◽

Vol 2025 (1) ◽

pp. 012050

Author(s):

Yuanmei Chen ◽

Haizhen Ren ◽

Lei Zhang ◽

Yang Zhao

Keyword(s):

Strong Product

Download Full-text

The Second Hyper-Zagreb Index of Complement Graphs and Its Applications of Some Nano Structures

Asian Journal of Probability and Statistics ◽

10.9734/ajpas/2021/v15i430364 ◽

2021 ◽

pp. 54-75

Author(s):

Mohammed Alsharafi ◽

Yusuf Zeren ◽

Abdu Alameri

Keyword(s):

Graph Theory ◽

Cartesian Product ◽

Quantitative Structure Activity Relationship ◽

Quantitative Structure ◽

Structure Property ◽

Zagreb Index ◽

Chemical Graph ◽

Strong Product ◽

Chemical Graph Theory ◽

Mathematical Operations

In chemical graph theory, a topological descriptor is a numerical quantity that is based on the chemical structure of underlying chemical compound. Topological indices play an important role in chemical graph theory especially in the quantitative structure-property relationship (QSPR) and quantitative structure-activity relationship (QSAR). In this paper, we present explicit formulae for some basic mathematical operations for the second hyper-Zagreb index of complement graph containing the join G1 + G2, tensor product G1 $\otimes$ G2, Cartesian product G1 x G2, composition G1 $\circ$ G2, strong product G1 * G2, disjunction G1 V G2 and symmetric difference G1 $\oplus$ G2. Moreover, we studied the second hyper-Zagreb index for some certain important physicochemical structures such as molecular complement graphs of V-Phenylenic Nanotube V PHX[q, p], V-Phenylenic Nanotorus V PHY [m, n] and Titania Nanotubes TiO2.

Download Full-text