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Journal articles on the topic 'Petersen graph'

Author: Grafiati

Published: 4 June 2021

Last updated: 25 July 2025

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1

P, Sumathi, and Suresh Kumar J. "Fuzzy Quotient -3 Cordial Labeling on Generalized Petersen Graph." Indian Journal of Science and Technology 16, no. 9 (2023): 648–59. https://doi.org/10.17485/IJST/v16i9.1720.

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Abstract <strong>Objectives:</strong> To analyze the existence of fuzzy quotient-3 cordial labeling on the generalized Petersen graph. <strong>Methods:</strong> The method involves mathematically defining how to label the vertex of a generalized Petersen graph and demonstrating that these formulations produce fuzzy quotient-3 cordial labeling. <strong>Findings:</strong> In this study, we proved that the generalized Petersen graph GP(h;m) ; 1 m < &lfloor; h 2 &rfloor; is fuzzy quotient-3 cordial, except for h 0( mod 6). <strong>Novelty:</strong> Here, we gi

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2

Ahmad, Mukhtar, Saddam Hussain, Ulfat Parveen, Iqra Zahid, Muhammad Sultan, and Ather Qayyum. "On Degree-Based Topological Indices of Petersen Subdivision Graph." European Journal of Mathematical Analysis 3 (June 5, 2023): 20. http://dx.doi.org/10.28924/ada/ma.3.20.

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In this paper, we adequately describe the generalised petersen graph, expanding to the categories of graphs. We created a petersen graph, which is cyclic and has vertices that are arranged in the centre and nine gons plus one vertex, leading to the factorization of regular graphs. Petersen graph is still shown in graph theory literature, nevertheless.

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3

Lv, Shengmei, and Liying Zhao. "Hamiltonian Indices of Three Classes of Graphs Obtained from Petersen Graph." Axioms 12, no. 6 (2023): 580. http://dx.doi.org/10.3390/axioms12060580.

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In this paper, we mainly consider the Hamiltonian indices of three classes of graphs obtained from Petersen graph, that is, the minimum integer m of m-time iterated line graph Lm(G) of these three classes of graphs such that Lm(G) is Hamiltonian. We show that the Hamiltonian indices of those graphs obtained by replacing every vertex of Petersen graph with a n-cycle or a complete graph of order n, or adding n pendant edges to each vertex of Petersen graph are both 2. In addition, we also study the situations of adding an edge to these three classes of graphs and obtain that their Hamiltonian in

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4

Beaula, C., та P. Venugopal. "Cryptosystem Using the Generalized Petersen Graph 𝐺𝑃 (𝟐𝑛 + 𝟏, 𝟐)". Indian Journal Of Science And Technology 17, № 30 (2024): 3125–37. http://dx.doi.org/10.17485/ijst/v17i30.1130.

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Objective: In the digital era, data transfer in a network without external interference is one of the challenging problems. External interference can be minimized by creating a strong cryptosystem. For this purpose, different mathematical concepts are incorporated to construct a cryptosystem. Recently, techniques in graph theory, a branch of mathematics, are also employed in cryptography. The objective of this paper is to propose a new cryptosystem to encrypt and decrypt an alphabetical string of lengths less than equal to 16 using graph decomposition and edge labeling on a generalized Peterse

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C, Beaula, та Venugopal P. "Cryptosystem Using the Generalized Petersen Graph 𝐺𝑃 (𝟐𝑛 + 𝟏, 𝟐)". Indian Journal of Science and Technology 17, № 30 (2024): 3125–37. https://doi.org/10.17485/IJST/v17i30.1130.

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Abstract <strong>Objective:</strong> In the digital era, data transfer in a network without external interference is one of the challenging problems. External interference can be minimized by creating a strong cryptosystem. For this purpose, different mathematical concepts are incorporated to construct a cryptosystem. Recently, techniques in graph theory, a branch of mathematics, are also employed in cryptography. The objective of this paper is to propose a new cryptosystem to encrypt and decrypt an alphabetical string of lengths less than equal to 16 using graph decomposition and edge la

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6

Li, Xiao Min, Lan Lei, Hong-Jian Lai, and Meng Zhang. "Supereulerian graphs and the Petersen graph." Acta Mathematica Sinica, English Series 30, no. 2 (2014): 291–304. http://dx.doi.org/10.1007/s10114-014-2272-y.

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7

Catlin, Paul A., and Hong-Jian Lai. "Supereulerian Graphs and the Petersen Graph." Journal of Combinatorial Theory, Series B 66, no. 1 (1996): 123–39. http://dx.doi.org/10.1006/jctb.1996.0009.

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8

Bednarz, Paweł, and Natalia Paja. "On (2-d)-Kernels in Two Generalizations of the Petersen Graph." Symmetry 13, no. 10 (2021): 1948. http://dx.doi.org/10.3390/sym13101948.

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A subset J is a (2-d)-kernel of a graph if J is independent and 2-dominating simultaneously. In this paper, we consider two different generalizations of the Petersen graph and we give complete characterizations of these graphs which have (2-d)-kernel. Moreover, we determine the number of (2-d)-kernels of these graphs as well as their lower and upper kernel number. The property that each of the considered generalizations of the Petersen graph has a symmetric structure is useful in finding (2-d)-kernels in these graphs.

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9

R, Suguna. "Characteristics of Fuzzy Petersen Graph and Platonic Graph with Fuzzy Rule." International Journal for Research in Applied Science and Engineering Technology 10, no. 2 (2022): 374–78. http://dx.doi.org/10.22214/ijraset.2022.40273.

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Abstract: Graph theory is the concepts used to study and model various application in different areas. In this paper, we consider the The Petersen and also the Platonic graph using if-then-rules fuzzy numbers. The results are related to the find the degree of odd vertices and even verticesare same by applying if-then-rules through the paths described by fuzzy numbers. Keywords: Degree of Vetex, Incident Graph, Peterson Graph, Platonic graph, Fuzzy IF-THEN rule.

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10

B.Jothilakshmi. "Degree-Based Topological Indices of Modified Petersen Graph." Panamerican Mathematical Journal 35, no. 4s (2025): 235–46. https://doi.org/10.52783/pmj.v35.i4s.4673.

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Objectives:The Petersen graph is non-planar, meaning it cannot be drawn on a plane without edge crossings. This limits its use in applications requiring planar graphs, such as circuit board design, geographic mapping, or any domain where planar embeddings are essential. The Petersen graph is fixed with only 10 vertices and 15 edges. Its small size can make it unsuitable for modelling or analysing larger, more complex systems. The weakness of unsuitable for modelling or analysing larger, more complex systems estimation is the lack of consideration of the graph and it is not a function of the ov

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11

Iqbal, Tanveer, Syed Ahtsham Ul Haq Bokhary, Shreefa O. Hilali, Mohammed Alhagyan, Ameni Gargouri, and Muhammad Naeem Azhar. "Edge Resolvability in Generalized Petersen Graphs." Symmetry 15, no. 9 (2023): 1633. http://dx.doi.org/10.3390/sym15091633.

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The generalized Petersen graphs are a type of cubic graph formed by connecting the vertices of a regular polygon to the corresponding vertices of a star polygon. This graph has many interesting graph properties. As a result, it has been widely researched. In this work, the edge metric dimensions of the generalized Petersen graphs GP(2l + 1, l) and GP(2l, l) are explored, and it is shown that the edge metric dimension of GP(2l + 1, l)) is equal to its metric dimension. Furthermore, it is proved that the upper bound of the edge metric dimension is the same as the value of the metric dimension fo

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12

Anderson, Ian, D. A. Holton, and J. Sheehan. "The Petersen Graph." Mathematical Gazette 79, no. 484 (1995): 239. http://dx.doi.org/10.2307/3620120.

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13

Irawan, Agus, and Ana Istiani. "The Locating Chromatic Number for the New Operation on Generalized Petersen Graphs N_P(m,1)." Sainmatika: Jurnal Ilmiah Matematika dan Ilmu Pengetahuan Alam 21, no. 1 (2024): 89–96. http://dx.doi.org/10.31851/sainmatika.v21i1.14864.

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The locating chromatic number is a graph invariant that quantifies the minimum number of colors required for proper vertex coloring, ensuring that any two vertices with the same color have distinct sets of neighbors. This study introduces a new operation on generalized Petersen graphs denoted by N_(P(m,1)), exploring its impact on locating chromatic numbers. Through systematic analysis, we aim to determine the specific conditions under which this operation influences the locating chromatic number and provide insights into the underlying graph-theoretical properties. The method for computing th

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14

Tamanna, Kumari, Kumar Kamal, and Agarwal Richa. "A STUDY OF STRUCTURE OF GRAPH INVARIANT WITH THEIR INDEPENDENCE NUMBERS AND CAYLEY GRAPHS." Soch – Mastnath Journal Of Science & Technology 17, no. 2 (2022): 1–8. https://doi.org/10.5281/zenodo.8009349.

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ABSTRACT In this paper the independence numbers and algebraic properties of graph invariant has been discussed and studied. The graph invariant is generalization of Petersen graphs whose independence numbers have been previously known. The authors studied the bounds for the independence number of different graph invariant and sub-classes of graph invariant, and exactly determine the independence number for other graph invariant and sub-classes of graph invariant. Authors also analysed the automorphism groups of the graph invariant.

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15

Harary, Frank, and Zsolt Tuza. "Two graph-colouring games." Bulletin of the Australian Mathematical Society 48, no. 1 (1993): 141–49. http://dx.doi.org/10.1017/s0004972700015549.

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16

Wang, Jinhua, and Dengju Ma. "Petersen Graph Decompositions of Complete Multipartite Graphs." Graphs and Combinatorics 26, no. 5 (2010): 737–44. http://dx.doi.org/10.1007/s00373-010-0925-x.

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17

Elzinga, Randall J., David A. Gregory, and Kevin N. Vander Meulen. "Addressing the Petersen graph." Electronic Notes in Discrete Mathematics 11 (July 2002): 278–83. http://dx.doi.org/10.1016/s1571-0653(04)00071-x.

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18

Chartrand, Gary, Héctor Hevia, and Robin J. Wilson. "The ubiquitous Petersen graph." Discrete Mathematics 100, no. 1-3 (1992): 303–11. http://dx.doi.org/10.1016/0012-365x(92)90649-z.

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19

Elzinga, Randall J., David A. Gregory, and Kevin N. Vander Meulen. "Addressing the Petersen graph." Discrete Mathematics 286, no. 3 (2004): 241–44. http://dx.doi.org/10.1016/j.disc.2004.06.006.

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20

Guirao, Juan, Sarfraz Ahmad, Muhammad Siddiqui, and Muhammad Ibrahim. "Edge Irregular Reflexive Labeling for Disjoint Union of Generalized Petersen Graph." Mathematics 6, no. 12 (2018): 304. http://dx.doi.org/10.3390/math6120304.

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A graph labeling is the task of integers, generally spoken to by whole numbers, to the edges or vertices, or both of a graph. Formally, given a graph G = ( V , E ) a vertex labeling is a capacity from V to an arrangement of integers. A graph with such a capacity characterized is known as a vertex-labeled graph. Similarly, an edge labeling is an element of E to an arrangement of labels. For this situation, the graph is called an edge-labeled graph. We examine an edge irregular reflexive k-labeling for the disjoint association of the cycle related graphs and decide the correct estimation of the

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21

Sim, Kai An, and Kok Bin Wong. "On the cooling number of the generalized Petersen graphs." AIMS Mathematics 9, no. 12 (2024): 36351–70. https://doi.org/10.3934/math.20241724.

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<p>Recently, Bonato et al. (2024) introduced a new graph parameter, which is the cooling number of a graph $ G $, denoted as CL$ (G) $. In contrast with burning which seeks to minimize the number of rounds to burn all vertices in a graph, cooling seeks to maximize the number of rounds to cool all vertices in the graph. Cooling number is the compelling counterpart to the well-studied burning number, offering a new perspective on dynamic processes within graphs. In this paper, we showed that the cooling number of a classic cubic graph, the generalized Petersen graphs $ P(n, k) $, is $ \lef

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22

BRANDT, STEPHAN. "On the Structure of Dense Triangle-Free Graphs." Combinatorics, Probability and Computing 8, no. 3 (1999): 237–45. http://dx.doi.org/10.1017/s0963548399003831.

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As a consequence of an early result of Pach we show that every maximal triangle-free graph is either homomorphic with a member of a specific infinite sequence of graphs or contains the Petersen graph minus one vertex as a subgraph. From this result and further structural observations we derive that, if a (not necessarily maximal) triangle-free graph of order n has minimum degree δ[ges ]n/3, then the graph is either homomorphic with a member of the indicated family or contains the Petersen graph with one edge contracted. As a corollary we get a recent result due to Chen, Jin and Koh. Finally, w

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23

Zhao, Weidong, Muhammad Naeem, and Irfan Ahmad. "Prime Cordial Labeling of Generalized Petersen Graph under Some Graph Operations." Symmetry 14, no. 4 (2022): 732. http://dx.doi.org/10.3390/sym14040732.

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A graph is a connection of objects. These objects are often known as vertices or nodes and the connection or relation in these nodes are called arcs or edges. There are certain rules to allocate values to these vertices and edges. This allocation of values to vertices or edges is called graph labeling. Labeling is prime cordial if vertices have allocated values from 1 to the order of graph and edges have allocated values 0 or 1 on a certain pattern. That is, an edge has an allocated value of 0 if the incident vertices have a greatest common divisor (gcd) greater than 1. An edge has an allocate

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24

Susilawati, Susilawati, Iis Nasfianti, Efni Agustiarini, and Dinda Khairani Nasution. "The Hamiltonian and Hypohamiltonian of Generalized Petersen Graph (GP_(n,9))." Jambura Journal of Mathematics 7, no. 1 (2025): 57–63. https://doi.org/10.37905/jjom.v7i1.30053.

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The study of Hamiltonian and Hypohamiltonian properties in the generalized Petersen graph GP_{n,k} is interesting due to the unique structure and characteristics of these graphs. The method employed in this study involves searching for Hamiltonian cycles within the generalized Petersen graph GP_{n,9}. Not all of GP_{n,9} graphs are Hamiltonian. For certain values of n, if the graph does not contain a Hamiltonian cycle, then one vertex should be removed from the graph to become Hamiltonian or neither. This research specifically investigates the Hypohamiltonian property of GP_{n,9}. The results

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25

M., Vasuki, and Dinesh Kumar A. "GRAPH THEORY WITH THEIR REAL-TIME APPLICATIONS IN EVERYDAY LIFE AND TECHNOLOGY." International Journal of Engineering Research and Modern Education (IJERME) 6, no. 1 (2021): 17–20. https://doi.org/10.5281/zenodo.6079366.

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A petal graph is a connected graph G with maximum degree three, minimum degree two, and such that the set of vertices of degree three induces a 2–regular graph and the whole set of vertices of degree in two induces an empty graph. We prove here that, with the single exception of the graph obtained from the Petersen graph by deleting one vertex, all petal graphs are Class 1.

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26

Chen, Min, and André Raspaud. "Homomorphisms from sparse graphs to the Petersen graph." Discrete Mathematics 310, no. 20 (2010): 2705–13. http://dx.doi.org/10.1016/j.disc.2010.04.002.

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27

Sun, Minhong, and Zehui Shao. "On Double Domination Numbers of Generalized Petersen Graphs." Journal of Computational and Theoretical Nanoscience 13, no. 10 (2016): 6514–18. http://dx.doi.org/10.1166/jctn.2016.5595.

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A (total) double dominating set in a graph G is a subset S ⊆ V(G) such that each vertex in V(G) is (total) dominated by at least 2 vertices in S. The (total) double domination number of G is the minimum size of a (total) double dominating set of G. We determine the total double domination numbers and give upper bounds for double domination numbers of generalized Petersen graphs. By applying an integer programming model for double domination numbers of a graph, we have determined some exact values of double domination numbers of these generalized Petersen graphs with small parameters. The resul

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Akwu, Abolape Deborah, and Deborah Olayide A. Ajayi. "Totally antimagic total labeling of ladders, prisms and generalised Petersen graphs." Journal of Discrete Mathematical Sciences & Cryptography 27, no. 1 (2024): 31–44. http://dx.doi.org/10.47974/jdmsc-1229.

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A total labeling is called edge-antimagic total (vertex-antimagic total) if all edge-weights (vertex-weights) are pairwise distinct. If a labeling is simultaneously edge-antimagic total and vertex-antimagic total, it is called a totally antimagic total labeling. A graph that admits totally antimagic total labeling is called a totally antimagic total graph. In this paper, we prove that ladders, prisms and generalised Pertersen graphs are totally antimagic total graphs. We also show that the chain graph of totally antimagic total graphs is a totally antimagic total graphs.

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Kang, Yingli, Shuai Ye, Weidong Fu, and Jing Zhu. "The Pessimistic Diagnosability of Folded Petersen Cubes." Journal of Mathematics 2022 (October 20, 2022): 1–9. http://dx.doi.org/10.1155/2022/3114022.

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Diagnosability is an important metric parameter for measuring the reliability of multiprocessor systems. The pessimistic diagnosis strategy is a classic diagnostic model based on the PMC model. The class of folded Petersen cubes, denoted by FP Q n , k , where n , k ≥ 0 and n , k ≠ 0 , 0 , is introduced as a competitive model of the hypercubes, which is constructed by iteratively applying the Cartesian product operation on the hypercube Q n and the Petersen graph P . In this paper, by exploring the structural properties of the folded Petersen cubes FP Q n , k , we first prove that FP Q n , k is

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30

Chen, Ming, Lianying Miao, and Shan Zhou. "Strong Edge Coloring of Generalized Petersen Graphs." Mathematics 8, no. 8 (2020): 1265. http://dx.doi.org/10.3390/math8081265.

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A strong edge coloring of a graph G is a proper edge coloring such that every color class is an induced matching. In 2018, Yang and Wu proposed a conjecture that every generalized Petersen graph P(n,k) with k≥4 and n>2k can be strong edge colored with (at most) seven colors. Although the generalized Petersen graph P(n,k) is a kind of special graph, the strong chromatic index of P(n,k) is still unknown. In this paper, we support the conjecture by showing that the strong chromatic index of every generalized Petersen graph P(n,k) with k≥4 and n>2k is at most 9.

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31

Sharma, Arti, and Atul Gaur. "Maximal labeling of graphs." Discrete Mathematics, Algorithms and Applications 11, no. 04 (2019): 1950044. http://dx.doi.org/10.1142/s1793830919500447.

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The notion of maximal labeling, optimal maximal labeling and the maximal index of a graph using the nonunit elements of a commutative ring with identity are introduced and studied. The maximal index of complete graphs, complete bipartite graphs are given. Maximal index of cycles of order up to [Formula: see text] and Petersen graph are also given.

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32

SHERMAN, DAVID, MING TSAI, CHENG-KUAN LIN, et al. "4-ORDERED HAMILTONICITY FOR SOME CHORDAL RING GRAPHS." Journal of Interconnection Networks 11, no. 03n04 (2010): 157–74. http://dx.doi.org/10.1142/s0219265910002787.

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A graph G is k-ordered if for any sequence of k distinct vertices of G, there exists a cycle in G containing these k vertices in the specified order. It is k-ordered Hamiltonian if, in addition, the required cycle is Hamiltonian. The question of the existence of an infinite class of 3-regular 4-ordered Hamiltonian graphs was posed in 1997 by Ng and Schultz.13At the time, the only known examples were K4and K3,3. Some progress was made in 2008 by Mészáros,12when the Peterson graph was found to be 4-ordered and the Heawood graph was proved to be 4-ordered Hamiltonian; moreover, an infinite class

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33

Ge, Huifen, Shumin Zhang, Chengfu Ye, and Rongxia Hao. "The generalized 4-connectivity of folded Petersen cube networks." AIMS Mathematics 7, no. 8 (2022): 14718–37. http://dx.doi.org/10.3934/math.2022809.

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<abstract><p>The generalized $ \ell $-connectivity $ \kappa_{\ell}(G) $ of a graph $ G $ is a generalization of classical connectivity $ \kappa(G) $ with $ \kappa_{2}(G) = \kappa(G) $. It serves to measure the capability of connection for any $ \ell $ vertices. The folded Petersen cube network $ FPQ_{n, k} $ can be used to model the topological structure of a communication-efficient multiprocessor. This paper shows that the generalized 4-connectivity of the folded Petersen cube network $ FPQ_{n, k} $ is $ n+3k-1 $. As a corollary, the generalized 3-connectivity of $ FPQ_{n, k} $ al

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34

Muhammad Rafiullah, Muhammad, Dur-E. Jabeen, and Mohamad Nazri Husin. "Some Mathematical Properties of Sombor Indices for Regular Graphs." Malaysian Journal of Fundamental and Applied Sciences 20, no. 6 (2024): 1392–97. https://doi.org/10.11113/mjfas.v20n6.3839.

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In 2021, Gutman introduced Sombor index (SO) of a graph G and defined as SO(G)=∑_(uv∈E(G))▒√(〖deg⁡(u)〗^2+deg(v)^2 ) . In this paper, we have calculated the Sombor index of r-regular graph G_r, line graph of G_r, 〖L(G〗_r) and complement graph of G_r, (G_r ) ̅. We have also discussed a particular case of regular graphs, generalized Petersen graph P(s,t), its line graph L((P(s,t)) and complement graph (P(s,t)) ̅ for s>4 . We have proved the relation between these graphs and categorized them on the base of the Sombor index.

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35

Kilakos, K., and B. Shepherd. "Excluding Minors in Cubic Graphs." Combinatorics, Probability and Computing 5, no. 1 (1996): 57–78. http://dx.doi.org/10.1017/s0963548300001838.

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Let P10\e be the graph obtained by deleting an edge from the Petersen graph. We give a decomposition theorem for cubic graphs with no minor isomorphic to P10\e. The decomposition is used to show that graphs in this class are 3-edge-colourable. We also consider an application to a conjecture due to Grötzsch which states that a planar graph is 3-edge-colourable if and only if it is fractionally 3-edge-colourable.

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36

McKay, Brendan D., and Cheryl E. Praeger. "Vertex-transitive graphs which are not Cayley graphs, I." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 56, no. 1 (1994): 53–63. http://dx.doi.org/10.1017/s144678870003473x.

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AbstractThe Petersen graph on 10 vertices is the smallest example of a vertex-transitive graph which is not a Cayley graph. We consider the problem of determining the orders of such graphs. In this, the first of a series of papers, we present a sequence of constructions which solve the problem for many orders. In particular, such graphs exist for all orders divisible by a fourth power, and all even orders which are divisible by a square.

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37

Shaheen, Ramy, Ziad Kanaya, and Khaled Alshehada. "Game Chromatic Number of Generalized Petersen Graphs and Jahangir Graphs." Journal of Applied Mathematics 2020 (April 14, 2020): 1–4. http://dx.doi.org/10.1155/2020/6475427.

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Let G = V , E be a graph, and two players Alice and Bob alternate turns coloring the vertices of the graph G a proper coloring where no two adjacent vertices are signed with the same color. Alice's goal is to color the set of vertices using the minimum number of colors, which is called game chromatic number and is denoted by χ g G , while Bob's goal is to prevent Alice's goal. In this paper, we investigate the game chromatic number χ g G of Generalized Petersen Graphs G P n , k for k ≥ 3 and arbitrary n , n -Crossed Prism Graph, and Jahangir Graph J n , m .

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38

Girish, Lakshmi, and Kanagasabapathi Somasundaram. "Bound for the k-Fault-Tolerant Power-Domination Number." Symmetry 16, no. 7 (2024): 781. http://dx.doi.org/10.3390/sym16070781.

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A set S⊆V is referred to as a k-fault-tolerant power-dominating set of a given graph G=(V,E) if the difference S∖F remains a power-dominating set of G for any F⊆S with |F|≤k, where k is an integer with 0≤k<|V|. The lowest cardinality of a k-fault-tolerant power-dominating set is the k-fault-tolerant power-domination number of G, denoted by γPk(G). Generalized Petersen graphs GP(m,k) and generalized cylinders SG are two well-known graph classes. In this paper, we calculate the k-fault-tolerant power-domination number of the generalized Petersen graphs GP(m,1) and GP(m,2). Also, we obtain γPk

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39

Iqbal, Tanveer, Muhammad Rafiq, Muhammad Naeem Azhar, Muhammad Salman, and Imran Khalid. "On the Edge Resolvability of Double Generalized Petersen Graphs." Journal of Mathematics 2022 (April 22, 2022): 1–14. http://dx.doi.org/10.1155/2022/6490698.

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For a connected graph G = V G , E G , let v ∈ V G be a vertex and e = uw ∈ E G be an edge. The distance between the vertex v and the edge e is given by d G e , v = min d G u , v , d G w , v . A vertex w ∈ V G distinguishes two edges e 1 , e 2 ∈ E G if d G w , e 1 ≠ d G w , e 2 . A well-known graph invariant related to resolvability of graph edges, namely, the edge resolving set, is studied for a family of 3 -regular graphs. A set S of vertices in a connected graph G is an edge metric generator for G if every two edges of G are distinguished by some vertex of S . The smallest cardinality of an

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40

Nasir, Sumiya, Nazeran Idrees, Afshan Sadiq, Fozia Bashir Farooq, Salma Kanwal, and Muhammad Imran. "Strongly Multiplicative Labeling of Diamond Graph, Generalized Petersen Graph, and Some Other Graphs." Journal of Mathematics 2022 (July 5, 2022): 1–5. http://dx.doi.org/10.1155/2022/3203108.

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A finite, simple graph of order k is said to be a strongly multiplicative graph when all vertices of the graph are labeled by positive integers 1,2,3 , … , k such that the induced edge labels of the graph, obtained by the product of labels of end vertices of edges, are distinct. In this paper, we show that the diamond graph B r n for 𝑛 ≥ 3, umbrella graph U m , n , and generalized Petersen graph GP n , k , for n ≥ 3 and 1 ≤ k < n / 2 , admit strongly multiplicative labeling. Moreover, strongly multiplicative labeling of a double comb graph and sunflower planar graph has also been investigat

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Zhang, Zhiqiang, Muhammad Naeem, Abeera Tariq, and Weidong Zhao. "On Square Sum Labeling of Two Families of Petersen Graphs." Journal of Mathematics 2022 (February 18, 2022): 1–14. http://dx.doi.org/10.1155/2022/1872695.

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A labeling on a graph G with n vertices and m edges is called square sum if there exists a bijection f : V G ⟶ 0,1,2,3 , … , n − 1 such that the function f ∗ : E G ⟶ N defined by f ∗ s t = f s 2 + f t 2 , for all s t ∈ E G , is injective. A graph G having a square sum labeling is called square sum. In this study, we have investigated the square sum labeling of generalized Petersen graph and double generalized Petersen graph.

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Angaleeswari, K., K. Krishnan, M. Perumal, and V. Swaminathan. "k-Extensibility and Weakly k-Extensibility in Generalized Petersen Graphs." Shanlax International Journal of Arts, Science and Humanities 8, S1-May (2021): 59–67. http://dx.doi.org/10.34293/sijash.v8is1-may.4509.

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Let G be a simple graph. Let k be a positive integer. G is said to be k- extendable if every independent set of cardinality k is contained in a maximum independent set of G. A graph is weakly k- extendable if any non-maximal independent set of cardinality k is contained in a maximal independent set of G. Every k- extendable is weakly k- extendable but not the converse. Thus weakly k- extendable graph is a class of graphs wider than the class of k-extendable graphs. k-extendable and weakly k- extendable have been studied in [1, 2,3,4,6]. Characterization of graphs with β0 (G) = (n - 3), β0 (G)

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Agustin, Ika Hesti, D. Dafik, and A. Y. Harsya. "On r-dynamic coloring of some graph operations." Indonesian Journal of Combinatorics 1, no. 1 (2016): 22. http://dx.doi.org/10.19184/ijc.2016.1.1.3.

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Let $G$ be a simple, connected and undirected graph. Let $r,k$ be natural number. By a proper $k$-coloring of a graph $G$, we mean a map $ c : V (G) \rightarrow S$, where $|S| = k$, such that any two adjacent vertices receive different colors. An $r$-dynamic $k$-coloring is a proper $k$-coloring $c$ of $G$ such that $|c(N (v))| \geq min\{r, d(v)\}$ for each vertex $v$ in $V(G)$, where $N (v)$ is the neighborhood of $v$ and $c(S) = \{c(v) : v \in S\}$ for a vertex subset $S$ . The $r$-dynamic chromatic number, written as $\chi_r(G)$, is the minimum $k$ such that $G$ has an $r$-dynamic $k$-color

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Shao, Zehui, Rija Erveš, Huiqin Jiang, Aljoša Peperko, Pu Wu, and Janez Žerovnik. "Double Roman Graphs in P(3k, k)." Mathematics 9, no. 4 (2021): 336. http://dx.doi.org/10.3390/math9040336.

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A double Roman dominating function on a graph G=(V,E) is a function f:V→{0,1,2,3} with the properties that if f(u)=0, then vertex u is adjacent to at least one vertex assigned 3 or at least two vertices assigned 2, and if f(u)=1, then vertex u is adjacent to at least one vertex assigned 2 or 3. The weight of f equals w(f)=∑v∈Vf(v). The double Roman domination number γdR(G) of a graph G is the minimum weight of a double Roman dominating function of G. A graph is said to be double Roman if γdR(G)=3γ(G), where γ(G) is the domination number of G. We obtain the sharp lower bound of the double Roman

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Saleem, H., M. N. Husin, S. Ali, and M. S. Hameed. "On Examining Metric Dimension Through Edge Contraction in Certain Families of Graphs." Malaysian Journal of Mathematical Sciences 19, no. 2 (2025): 613–35. https://doi.org/10.47836/mjms.19.2.13.

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In graph theory, the Metric Dimension (MD) is an elementary metric that affords evidence nearly the essential selves of graphs. We reconnoiter the MD in the venue of edge-contracted regular graphs in this paper, with exceptional devotion to the Antiprism, Petersen, and Harary graphs. Our effort creates a vital bond between antiprism and its edge-contracted counterpart: we give a scheme to regulate the MD of the edge-contracted graph, given the MD of the novel graph. We likewise inspect how edge contraction affects regular graphs’ MDs, providing insight into how this operation deviations both t

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LI, XIAOJUAN, BING WEI, and YONGJIN ZHU. "CYCLES IN TRIANGLE-FREE GRAPHS." Discrete Mathematics, Algorithms and Applications 03, no. 03 (2011): 343–56. http://dx.doi.org/10.1142/s1793830911001267.

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Let G be a k-connected (k ≥ 3), triangle-free graph with α(G) ≤ k + 1. If G is not Petersen graph and G ∉ {Kk, k, Kk, k + 1, Kk + 1, k+1}, then G contains cycles of lengths from 4 to |V(G)|. This generalizes a result conjectured by Amar et al. (Graphs Combin.7 (1991)) and proved by Lou (Discrete Math.152 (1996)).

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Kratica, Jozef, Vera Kovacevic-Vujcic, and Mirjana Cangalovic. "K-metric antidimension of some generalized Petersen graphs." Filomat 33, no. 13 (2019): 4085–93. http://dx.doi.org/10.2298/fil1913085k.

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Resistance of social graphs to active attacks is a very important feature which must be maintained in the modern networks. Recently introduced k-metric antidimension graph invariant is used to define anew measure for resistance of social graphs. In this paper we have found and proved the k-metric antidimension for generalized Petersen graphs GP(n, 1) and GP(n,2). It is proven that GP(2m+1,1) and GP(8,2) are 2-metric antidimensional, while all other GP(n,1) and GP(n,2) graphs are 3-metric antidimensional.

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Shaheen, Ramy, Suhail Mahfud, and Qays Alhawat. "Hosoya, Schultz, and Gutman Polynomials of Generalized Petersen Graphs P n , 1 and P n , 2." Journal of Mathematics 2023 (July 3, 2023): 1–18. http://dx.doi.org/10.1155/2023/7341285.

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The graph theory has wide important applications in various other types of sciences. In chemical graph theory, we have many topological polynomials for a graph G through which we can compute many topological indices. Topological indices are numerical values and descriptors which are used to quantify the physiochemical properties and bioactivities of the chemical graph. In this paper, we compute Hosoya polynomial, hyper-Wiener index, Tratch–Stankevitch–Zefirov index, Harary index, Schultz polynomial, Gutman polynomial, Schultz index, and Gutman index of generalized Petersen graphs P n , 1 and P

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Rupnik Poklukar, Darja, and Janez Žerovnik. "On the Double Roman Domination in Generalized Petersen Graphs P(5k,k)." Mathematics 10, no. 1 (2022): 119. http://dx.doi.org/10.3390/math10010119.

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A double Roman dominating function on a graph G=(V,E) is a function f:V→{0,1,2,3} satisfying the condition that every vertex u for which f(u)=0 is adjacent to at least one vertex assigned 3 or at least two vertices assigned 2, and every vertex u with f(u)=1 is adjacent to at least one vertex assigned 2 or 3. The weight of f equals w(f)=∑v∈Vf(v). The double Roman domination number γdR(G) of a graph G equals the minimum weight of a double Roman dominating function of G. We obtain closed expressions for the double Roman domination number of generalized Petersen graphs P(5k,k). It is proven that γ

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G. Navamani. "Optimizing Subdivisions on Fair Domination in Petersen Graph Structures." Advances in Nonlinear Variational Inequalities 28, no. 3s (2024): 20–29. https://doi.org/10.52783/anvi.v28.2845.

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A fair dominating set (FDS) J is a dominating set of a graph H, such that all vertices in V \J are dominated by the same number of vertices of J. The fair domination number (FDN) is the minimum cardinality of a fair dominating set of H, denoted by γ_fd (H). The domination subdivision number (FDSN) 〖Sd〗_γ (H), represents the smallest number of edges in H that needs to be subdivided (with each edge being subdivided at most once) to increase the domination number of the graph. The fair domination subdivision number, 〖Sd〗_(γ_fd)^+ (H) (or 〖Sd〗_(γ_fd)^- (H)) is the minimum number of edge subdivisio

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