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3. L(4, 3, 2, 1)-Labeling for Simple GraphsSoumen Atta, Priya Ranjan Sinha Mahapatra, 2015, published scientific conference contribution Abstract: An L(4, 3, 2, 1)-labeling of a graph is a function which assigns label to each vertex of the graph such that if two vertices are one, two, three and four distance apart then assigned labels must have a difference of at least 4, 3, 2 and 1 respectively between them. This paper presents L(4, 3, 2, 1)-labeling number for simple graphs such as complete graphs, complete bipartite graphs, stars, paths and cycles. This paper also presents an L(4, 3, 2, 1)-labeling algorithm for paths which is optimal for paths on n≥7 vertices.
Keywords: L(4, 3, 2, 1)-labeling, Labeling number, Graph labeling, Channel assignment problem
Published in RUNG: 05.06.2023; Views: 692; Downloads: 0
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4. L (D, 2, 1)-labeling of Square GridSoumen Atta, Priya Ranjan Sinha Mahapatra, 2019, original scientific article Abstract: For a fixed integer $D (\geq 3)$ and $\lambda$ $\in$ $\mathbb{Z}^+$, a $\lambda$-$L(D, 2, 1)$-$labeling$ of a graph $G = (V, E)$ is the problem of assigning non-negative integers (known as labels) from the set $\{0, \ldots, \lambda\}$ to the vertices of $G$ such that if any two vertices in $V$ are one, two and three distance apart from each other then the assigned labels to these vertices must have a difference of at least $D$, 2 and 1 respectively. The vertices which are at least $4$ distance apart can receive the same label. The minimum value among all the possible values of $\lambda$ for which there exists a $\lambda$-$L(D, 2, 1)$-$labeling$ is known as the labeling number. In this paper $\lambda$-$L(D, 2 ,1)$-$labeling$ of square grid is considered. The lower bound on the labeling number for square grid is presented and a formula for $\lambda$-$L(D, 2 ,1)$-$labeling$ of square grid is proposed. The correctness proof of the proposed formula is given here. The upper bound of the labeling number obtained from the proposed labeling formula for square grid matches exactly with the lower bound of the labeling number.
Keywords: Graph labeling, Square grid, Labeling number, Frequency assignment problem (FAP)
Published in RUNG: 17.04.2023; Views: 740; Downloads: 0
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5. No-hole λ-L (k, k – 1, …, 2,1)-labeling for square gridSoumen Atta, Stanisław Goldstein, Priya Ranjan Sinha Mahapatra, 2017, original scientific article Abstract: Motivated by a frequency assignment problem, we demonstrate, for a fixed positive integer k, how to label an infinite square grid with a possibly small number of integer labels, ranging from 0 to λ −1, in such a way that labels of adjacent vertices differ by at least k, vertices connected by a path of length two receive values which differ by at least k − 1, and so on. The vertices which are at least k + 1 distance apart may receive the same label. By finding a lower bound for λ, we prove that the solution is close to optimal, with approximation ratio at most 9/8. The labeling presented is a no-hole one, i.e., it uses each of the allowed labels at least once.
Keywords: graph labeling, labeling number, no-hole labeling, square grid, frequency assignment problem, approximation ratio
Published in RUNG: 17.04.2023; Views: 725; Downloads: 0
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7. Semihypergroup-Based Graph for Modeling International Spread of COVID-n in Social SystemsNarjes Firouzkouhi, Reza Ameri, Abbas Amini, Hashem Bordbar, 2022, original scientific article Abstract: Graph theoretic techniques have been widely applied to model many types of links in social systems. Also, algebraic hypercompositional structure theory has demonstrated its systematic
application in some problems. Influenced by these mathematical notions, a novel semihypergroup based graph (SBG) of G = hH, Ei is constructed through the fundamental relation gn on H, where semihypergroup H is appointed as the set of vertices and E is addressed as the set of edges on SBG. Indeed, two arbitrary vertices x and y are adjacent if xgny. The connectivity of graph G is characterized by xg y, whereby the connected components SBG of G would be exactly the elements of the fundamental group H/g . Based on SBG, some fundamental characteristics of the graph such as complete, regular, Eulerian, isomorphism, and Cartesian products are discussed along with illustrative examples to clarify the relevance between semihypergroup H and its corresponding graph. Furthermore, the notions of geometric space, block, polygonal, and connected components are introduced in terms of the developed SBG. To formulate the links among individuals/countries in
the wake of the COVID (coronavirus disease) pandemic, a theoretical SBG methodology is presented to analyze and simplify such social systems. Finally, the developed SBG is used to model the trend diffusion of the viral disease COVID-n in social systems (i.e., countries and individuals).
Keywords: graph theory, hypergroup, fundamental relation, social systems, geometric space
Published in RUNG: 23.11.2022; Views: 990; Downloads: 0
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8. Bioacoustic signal analysis through complex network featuresMohanachandran Nair Sindhu Swapna, RAJ VIMAL, Sankararaman S, 2022, original scientific article Abstract: The paper proposes a graph-theoretical approach to auscultation, bringing out the potential of graph features in
classifying the bioacoustics signals. The complex network analysis of the bioacoustics signals - vesicular (VE) and
bronchial (BR) breath sound - of 48 healthy persons are carried out for understanding the airflow dynamics
during respiration. The VE and BR are classified by the machine learning techniques extracting the graph features
– the number of edges (E), graph density (D), transitivity (T), degree centrality (Dcg) and eigenvector centrality
(Ecg). The higher value of E, D, and T in BR indicates the temporally correlated airflow through the wider
tracheobronchial tract resulting in sustained high-intense low-frequencies. The frequency spread and high-frequencies in VE, arising due to the less correlated airflow through the narrow segmental bronchi and lobar,
appears as a lower value for E, D, and T. The lower values of Dcg and Ecg justify the inferences from the spectral
and other graph parameters. The study proposes a methodology in remote auscultation that can be employed in
the current scenario of COVID-19.
Keywords: Bioacoustic signal, Graph theory, Complex network, Lung auscultation
Published in RUNG: 30.06.2022; Views: 1199; Downloads: 0
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