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Journal articles on the topic 'Complement of graph'

Author: Grafiati

Published: 3 June 2025

Last updated: 31 July 2025

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1

Chellaram Malaravan, A., and A. Wilson Baskar. "A study on distance in graph complement." Discrete Mathematics, Algorithms and Applications 12, no. 03 (2020): 2050045. http://dx.doi.org/10.1142/s1793830920500457.

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The aim of this paper is to determine radius and diameter of graph complements. We provide a necessary and sufficient condition for the complement of a graph to be connected, and determine the components of graph complement. Finally, we completely characterize the class of graphs [Formula: see text] for which the subgraph induced by central (respectively peripheral) vertices of its complement in [Formula: see text] is isomorphic to a complete graph [Formula: see text], for some positive integer [Formula: see text].

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2

Wafiq, Hibi. "Non-Isomorphism Between Graph And Its Complement." Multicultural Education 7, no. 6 (2021): 256. https://doi.org/10.5281/zenodo.4965942.

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<em>It is known that any graph with six vertices cannot be isomorphic to its complement [3].V. K. Balakrishnan has written in his book Schaum’s solved problems series [1] the following: “Given two arbitrary Simple graphs of the same order and the same size, the problem of determining whetheran isomorphism exists between the two is known as the isomorphism problem in graph theory. In general, itis not all easy (in other words, there is no "efficient algorithm") to solve an arbitrary instance of the isomorphismproblem”, from here came the idea of this paper. As mentio

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3

Narayan, Sivaram, and Yousra Sharawi. "BOUNDS ON THE SUM OF MINIMUM SEMIDEFINITE RANK OF A GRAPH AND ITS COMPLEMENT." Electronic Journal of Linear Algebra 34 (February 21, 2018): 399–406. http://dx.doi.org/10.13001/1081-3810.3539.

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The minimum semi-definite rank (msr) of a graph is the minimum rank among all positive semi-definite matrices associated to the graph. The graph complement conjecture gives an upper bound for the sum of the msr of a graph and the msr of its complement. It is shown that when the msr of a graph is equal to its independence number, the graph complement conjecture holds with a better upper bound. Several sufficient conditions are provided for the msr of different classes of graphs to equal to its independence number.

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4

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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5

Koch, Sebastian. "About Graph Complements." Formalized Mathematics 28, no. 1 (2020): 41–63. http://dx.doi.org/10.2478/forma-2020-0004.

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6

Kırcalı Gürsoy, Necla Kırcalı. "Graphs of Wajsberg Algebras via Complement Annihilating." Symmetry 15, no. 1 (2023): 121. http://dx.doi.org/10.3390/sym15010121.

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In this paper, W-graph, called the notion of graphs on Wajsberg algebras, is introduced such that the vertices of the graph are the elements of Wajsberg algebra and the edges are the association of two vertices. In addition to this, commutative W-graphs are also symmetric graphs. Moreover, a graph of equivalence classes of Wajsberg algebra is constructed. Meanwhile, new definitions as complement annihilator and ▵-connection operator on Wajsberg algebras are presented. Lemmas and theorems on these notions are proved, and some associated results depending on the graph’s algebraic properties are

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7

Visweswaran, S. "When Is the Complement of the Zero-Divisor Graph of a Commutative Ring Complemented?" ISRN Algebra 2012 (June 16, 2012): 1–13. http://dx.doi.org/10.5402/2012/282054.

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Let be a commutative ring with identity which has at least two nonzero zero-divisors. Suppose that the complement of the zero-divisor graph of has at least one edge. Under the above assumptions on , it is shown in this paper that the complement of the zero-divisor graph of is complemented if and only if is isomorphic to as rings. Moreover, if is not isomorphic to as rings, then, it is shown that in the complement of the zero-divisor graph of , either no vertex admits a complement or there are exactly two vertices which admit a complement.

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8

Elakkiya, A., and M. Yamuna. "Planar Characterization – Graph Domination Graphs." International Journal of Engineering & Technology 7, no. 4.10 (2018): 949. http://dx.doi.org/10.14419/ijet.v7i4.10.26634.

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9

AL-Omeri, Wadei Faris AL, M. Kaviyarasu, and Rajeshwari M. "Identifying Internet Streaming Services using Max Product of Complement in Neutrosophic Graphs." International Journal of Neutrosophic Science 23, no. 1 (2024): 257–72. http://dx.doi.org/10.54216/ijns.230123.

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The complement of the highest result of multiplication of two neutrosophic graphs is determined in this study. In the complement of the maximum product of a neutrosophic graph, the degree of a vertex is investigated. The complement of the maximum product of two normal neutrosophic graphs has several results that are presented and proven. Finally, we have offered a neutrosophic graph application for locating an online streaming service using normalized Hamming distance.

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10

Janakiraman, T. N., S. Muthammai, and M. Bhanumathi. "On the Boolean function graph of a graph and on its complement." Mathematica Bohemica 130, no. 2 (2005): 113–34. http://dx.doi.org/10.21136/mb.2005.134130.

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11

Mitchell, Lon. "Graph degeneracy and orthogonal vector representations." Electronic Journal of Linear Algebra 39 (May 18, 2023): 282–85. http://dx.doi.org/10.13001/ela.2023.6907.

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12

Visweswaran, S., and Hiren D. Patel. "Some results on the complement of the annihilating ideal graph of a commutative ring." Journal of Algebra and Its Applications 14, no. 07 (2015): 1550099. http://dx.doi.org/10.1142/s0219498815500991.

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Rings considered in this article are commutative with identity which admit at least one nonzero annihilating ideal. For such a ring R, we determine necessary and sufficient conditions in order that the complement of its annihilating ideal graph is connected and also find its diameter when it is connected. We discuss the girth of the complement of the annihilating ideal graph of R and prove that it is either equal to 3 or ∞. We also present a necessary and sufficient condition for the complement of the annihilating ideal graph to be complemented.

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13

Zelinka, Bohdan. "The distance between a graph and its complement." Czechoslovak Mathematical Journal 37, no. 1 (1987): 120–23. http://dx.doi.org/10.21136/cmj.1987.102139.

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14

Ramprasad, Ch, P. L. N. Varma, S. Satyanarayana, and N. Srinivasarao. "Vertex Degrees and Isomorphic Properties in Complement of an m-Polar Fuzzy Graph." Advances in Fuzzy Systems 2017 (2017): 1–9. http://dx.doi.org/10.1155/2017/3817469.

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Computational intelligence and computer science rely on graph theory to solve combinatorial problems. Normal product and tensor product of an m-polar fuzzy graph have been introduced in this article. Degrees of vertices in various product graphs, like Cartesian product, composition, tensor product, and normal product, have been computed. Complement and μ-complement of an m-polar fuzzy graph are defined and some properties are studied. An application of an m-polar fuzzy graph is also presented in this article.

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15

S. Raikar, Vidya, Shailaja S. Shirkol, and Preeti B. Jinagouda. "SECOND ZAGREB INDEX OF k-SPLITTING GRAPH OF GENERALIZED TRANSFORMATION GRAPHS." Journal of Dynamics and Control 9, no. 5 (2025): 57–73. https://doi.org/10.71058/jodac.v9i5006.

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Recently, in [18] Raikar et al. obtained explicit formulae for the first Zagreb index and coindex of k-splitting of generalized transformation graphs and their complement. The aim of this paper is to obtain explicit formulae for the second Zagreb index and coindex of k-splitting of generalized transformation graphs splk(Gab), and followed by this we also obtained analogous expressions for the complements of splk(Gab).

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DEVHARE, SARIKA, VINAYAK JOSHI, and JOHN LAGRANGE. "ON THE COMPLEMENT OF THE ZERO-DIVISOR GRAPH OF A PARTIALLY ORDERED SET." Bulletin of the Australian Mathematical Society 97, no. 2 (2017): 185–93. http://dx.doi.org/10.1017/s0004972717000867.

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In this paper, it is proved that the complement of the zero-divisor graph of a partially ordered set is weakly perfect if it has finite clique number, completely answering the question raised by Joshi and Khiste [‘Complement of the zero divisor graph of a lattice’,Bull. Aust. Math. Soc. 89(2014), 177–190]. As a consequence, the intersection graph of an intersection-closed family of nonempty subsets of a set is weakly perfect if it has finite clique number. These results are applied to annihilating-ideal graphs and intersection graphs of submodules.

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17

Kiggal Udayashankar, Kiran, and Prameela Kolake. "ON SELF COMPLEMENTARITY OF THE INDUCED COMPLEMENT OF A GRAPH." Sarajevo Journal of Mathematics 20, no. 2 (2025): 197–206. https://doi.org/10.5644/sjm.20.02.02.

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Let G = (V,E) be a graph and S ⊆V. The induced complement of the graph G with respect to the set S, denoted by GS, is the graph obtained from the graph G by removing the edges of ⟨S⟩ of G and adding the edges which are not in ⟨S⟩ of G. Given a set S ⊆V, the graph G is said to be S-induced self complementary if GS ∼= G. The graph G is said to be S-induced co-complementary if GS ∼= G. This paper presents the study of the different properties of the S-i.s.c. and S-i.c.c. graphs.

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18

Govindan, Vetrivel, Mullai Murugappan, Grienggrai Rajchakit, Surya R, and Subraja Saravanan. "Complement Properties of Pythagorean Co-Neutrosophic Graphs." International Journal on Robotics, Automation and Sciences 6, no. 2 (2024): 42–51. http://dx.doi.org/10.33093/ijoras.2024.6.2.7.

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The origination of graphs with neutrosophic type where membership of indeterminacy expels the vague results, by increasing the accuracy is used to extend application through the graphical environment. Since it is an extension of the intuitionistic type, there comes an immediate need to extend its findings and application to the neutrosophic type. Reversing the conditions of neutrosophic graphs by introducing the anti-behavior properties will produce an adequate number of new results and data, breaking the backlog in approaching decision-making problems and other real-world applications. This r

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19

Liu, Jia-Bao, Muhammad Javaid, Mohsin Raza, and Naeem Saleem. "On minimum algebraic connectivity of graphs whose complements are bicyclic." Open Mathematics 17, no. 1 (2019): 1490–502. http://dx.doi.org/10.1515/math-2019-0119.

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Abstract The second smallest eigenvalue of the Laplacian matrix of a graph (network) is called its algebraic connectivity which is used to diagnose Alzheimer’s disease, distinguish the group differences, measure the robustness, construct multiplex model, synchronize the stability, analyze the diffusion processes and find the connectivity of the graphs (networks). A connected graph containing two or three cycles is called a bicyclic graph if its number of edges is equal to its number of vertices plus one. In this paper, firstly the unique graph with a minimum algebraic connectivity is character

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20

Janakiraman, T. N., S. Muthammai, and M. Bhanumathi. "Domination numbers on the complement of the Boolean function graph of a graph." Mathematica Bohemica 130, no. 3 (2005): 247–63. http://dx.doi.org/10.21136/mb.2005.134098.

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21

Liu, Xiaogang, and Chengxin Yan. "Unitary homogeneous bi-Cayley graphs over finite commutative rings." Journal of Algebra and Its Applications 19, no. 09 (2019): 2050173. http://dx.doi.org/10.1142/s021949882050173x.

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Let [Formula: see text] denote the unitary homogeneous bi-Cayley graph over a finite commutative ring [Formula: see text]. In this paper, we determine the energy of [Formula: see text] and that of its complement and line graph, and characterize when such graphs are hyperenergetic. We also give a necessary and sufficient condition for [Formula: see text] (respectively, the complement of [Formula: see text], the line graph of [Formula: see text]) to be Ramanujan.

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22

Jansrang, Monsikarn, and Sivaram K. Narayan. "Graph complement conjecture for classes of shadow graphs." Operators and Matrices, no. 2 (2021): 589–614. http://dx.doi.org/10.7153/oam-2021-15-40.

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23

Abdul Razak, M. N. S., W. H. Fong, and N. H. Sarmin. "Graph splicing rules with cycle graph and its complement on complete graphs." Journal of Physics: Conference Series 1988, no. 1 (2021): 012067. http://dx.doi.org/10.1088/1742-6596/1988/1/012067.

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24

Tian, Yingzhi, Huaping Ma, and Liyun Wu. "The Connectivity of a Bipartite Graph and Its Bipartite Complementary Graph." Parallel Processing Letters 30, no. 03 (2020): 2040005. http://dx.doi.org/10.1142/s0129626420400058.

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In 1956, Nordhaus and Gaddum gave lower and upper bounds on the sum and the product of the chromatic number of a graph and its complement, in terms of the order of the graph. Since then, any bound on the sum and/or the product of an invariant in a graph [Formula: see text] and the same invariant in the complement [Formula: see text] of [Formula: see text] is called a Nordhaus-Gaddum type inequality or relation. The Nordhaus-Gaddum type inequalities for connectivity have been studied by several authors. For a bipartite graph [Formula: see text] with bipartition ([Formula: see text]), its bipart

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25

Chellaram Malaravan, A. "A study on center of a graph complement." Discrete Mathematics, Algorithms and Applications 07, no. 04 (2015): 1550046. http://dx.doi.org/10.1142/s1793830915500469.

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Characterization of class of graphs [Formula: see text] for which the complements have radius 2, and the subgraphs induced by central vertices of their complements isomorphic to a complete graph, is presented. In addition, the structure and the number of trees whose complements have subtrees induced by central vertices of their complements isomorphic to a complete graph of order [Formula: see text] with [Formula: see text] are determined.

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26

VISWESWARAN, S. "SOME RESULTS ON THE COMPLEMENT OF THE ZERO-DIVISOR GRAPH OF A COMMUTATIVE RING." Journal of Algebra and Its Applications 10, no. 03 (2011): 573–95. http://dx.doi.org/10.1142/s0219498811004781.

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Let R be a commutative ring with identity admitting at least two nonzero zero-divisors. First, in this article we determine when the complement of the zero-divisor graph of R is connected and also determine its diameter when it is connected. Second, in this article we study the relationship between the connectedness of the complement of the zero-divisor graph of R to that of the connectedness of the complement of the zero-divisor graph of T where either T = R[x] or T = R[[x]] and we study the relationship between their diameters in the case when both the graphs are connected. Finally, we give

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27

Bhat, K. Arathi, and G. Sudhakara. "Commuting decomposition of Kn1,n2,...,nk through realization of the product A(G)A(GPk )." Special Matrices 6, no. 1 (2018): 343–56. http://dx.doi.org/10.1515/spma-2018-0028.

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Abstract In this paper, we introduce the notion of perfect matching property for a k-partition of vertex set of given graph. We consider nontrivial graphs G and GPk , the k-complement of graph G with respect to a kpartition of V(G), to prove that A(G)A(GPk ) is realizable as a graph if and only if P satis_es perfect matching property. For A(G)A(GPk ) = A(Γ) for some graph Γ, we obtain graph parameters such as chromatic number, domination number etc., for those graphs and characterization of P is given for which GPk and Γ are isomorphic. Given a 1-factor graph G with 2n vertices, we propose a p

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28

Scaria, Deena C., John Joy Mulloor, Liju Alex, and Gopal Indulal. "A note on some graph parameters and graph operations." Open Journal of Discrete Applied Mathematics 8, no. 2 (2025): 32–44. https://doi.org/10.30538/psrp-odam2025.0114.

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This paper introduces the concept of the extended \(H\)-cover of a graph \(G\), denoted as \(G^*_H\) , as a generalization inspired by the extended double cover graphs discussed in Chen [1]. We explore the spectral properties and energy characteristics of \(G^*_H\), deriving formulae for the number of spanning trees in cases where both \(G\) and \(H\) are regular. Our investigation identifies several infinite families of equienergetic graphs and highlights instances of cospectral graphs within \(G^*_H\) . Additionally, we analyze various graph parameters related to the Indu-Bala product of gra

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29

Basavanagoud, B., and Veena Mathad. "Graph Equations for Line Graphs, Jump Graphs, Middle Graphs, Splitting Graphs And Line Splitting Graphs." Mapana - Journal of Sciences 9, no. 2 (2010): 53–61. http://dx.doi.org/10.12723/mjs.17.7.

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For a graph G, let G, L(G), J(G) S(G), L,(G) and M(G) denote Complement, Line graph, Jump graph, Splitting graph, Line splitting graph and Middle graph respectively. In this paper, we solve the graph equations L(G) =S(H), M(G) = S(H), L(G) = LS(H), M(G) =LS(H), J(G) = S(H), M(G) = S(H), J(G) = LS(H) and M(G) = LS(G). The equality symbol '=' stands for on isomorphism between two graphs.

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30

Daoub, Hamza, Osama Shafah, and Ahmad A. Almutlg. "Exploring a Graph Complement in Quadratic Congruence." Symmetry 16, no. 2 (2024): 213. http://dx.doi.org/10.3390/sym16020213.

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In this work, we investigate essential definitions, defining G as a simple graph with vertices in ℤn and subgraphs Γu and Γq as unit residue and quadratic residue graphs modulo n, respectively. The investigation extends to the degree of G, Γu, and Γq, illuminating the properties of these subgraphs in the context of quadratic congruences.

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31

Kostochka, Alexander, and B. Sudakov. "On Ramsey Numbers of Sparse Graphs." Combinatorics, Probability and Computing 12, no. 5-6 (2003): 627–41. http://dx.doi.org/10.1017/s0963548303005728.

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The Ramsey number, , of a graph G is the minimum integer N such that, in every 2-colouring of the edges of the complete graph on N vertices, there is a monochromatic copy of G. In 1975, Burr and Erdős posed a problem on Ramsey numbers of d-degenerate graphs, i.e., graphs in which every subgraph has a vertex of degree at most d. They conjectured that for every d there exists a constant c(d) such that for any d-degenerate graph G of order n.In this paper we prove that for each such G. In fact, we show that, for every , sufficiently large n, and any graph H of order , either H or its complement c

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DOBSON, EDWARD. "Constructing Trees in Graphs whose Complement has no K2,s." Combinatorics, Probability and Computing 11, no. 4 (2002): 343–47. http://dx.doi.org/10.1017/s0963548302005102.

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We show that, if G is a graph of order n with maximal degree Δ(G) and minimal degree δ(G) whose complement contains no K2,s, s [ges ] 2, then G contains every tree T of order n−s+1 whose maximal degree is at most Δ(G) and whose vertex of second-largest degree is at most δ(G). We then show that this result implies that special cases of two conjectures are true. We verify that the Erdös–Sós conjecture, which states that a graph whose average degree is larger than k−1 contains every tree of order k+1, is true for graphs whose complement does not contain a K2,4, and the Komlós–Sós conjecture, whic

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Fatimah, Farah Maulidya, Vira Hari Krisnawati, and Noor Hidayat. "Prime Graph over Cartesian Product over Rings and Its Complement." JTAM (Jurnal Teori dan Aplikasi Matematika) 7, no. 3 (2023): 712. http://dx.doi.org/10.31764/jtam.v7i3.14987.

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Graph theory is a branch of algebra that is growing rapidly both in concept and application studies. This graph application can be used in chemistry, transportation, cryptographic problems, coding theory, design communication network, etc. There is currently a bridge between graphs and algebra, especially an algebraic structures, namely theory of graph algebra. One of researchs on graph algebra is a graph that formed by prime ring elements or called prime graph over ring R. The prime graph over commutative ring R (PG(R))) is a graph construction with set of vertices V(PG(R))=R and two vertices

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Chandoor, Susanth, and Sunny Joseph Kalayathankal. "Operations on covering numbers of certain graph classes." International Journal of Advanced Mathematical Sciences 4, no. 1 (2016): 1. http://dx.doi.org/10.14419/ijams.v4i1.5531.

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<p><span>The bounds on the sum and product of chromatic numbers of a graph and its complement are known as Nordhaus-Gaddum inequalities. In a similar way, the operations on the covering numbers of graphs with their complement are studied and with respect to this, new characterizations of certain graph classes have also been given in this paper.</span></p>

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Pai, Amrithalakshmi, Harshitha A. Rao, Sabitha D’Souza, Pradeep G. Bhat та Shankar Upadhyay. "δ-Complement of a Graph". Mathematics 10, № 8 (2022): 1203. http://dx.doi.org/10.3390/math10081203.

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Let G(V,X) be a finite and simple graph of order n and size m. The complement of G, denoted by G¯, is the graph obtained by removing the lines of G and adding the lines that are not in G. A graph is self-complementary if and only if it is isomorphic to its complement. In this paper, we define δ-complement and δ′-complement of a graph as follows. For any two points u and v of G with degu=degv remove the lines between u and v in G and add the lines between u and v which are not in G. The graph thus obtained is called δ-complement of G. For any two points u and v of G with degu≠degv remove the li

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36

Mahato, Iswar, and M. Rajesh Kannan. "Extremal problems for the eccentricity matrices of complements of trees." Electronic Journal of Linear Algebra 39 (June 23, 2023): 339–54. http://dx.doi.org/10.13001/ela.2023.7781.

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The eccentricity matrix of a connected graph $G$, denoted by $\mathcal{E}(G)$, is obtained from the distance matrix of $G$ by keeping the largest nonzero entries in each row and each column and leaving zeros in the remaining ones. The $\mathcal{E}$-eigenvalues of $G$ are the eigenvalues of $\mathcal{E}(G)$. The largest modulus of an eigenvalue is the $\mathcal{E}$-spectral radius of $G$. The $\mathcal{E}$-energy of $G$ is the sum of the absolute values of all $\mathcal{E}$-eigenvalues of $G$. In this article, we study some of the extremal problems for eccentricity matrices of complements of tr

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Huang, Liangsong, Yu Hu, Yuxia Li, P. K. Kishore Kumar, Dipak Koley, and Arindam Dey. "A Study of Regular and Irregular Neutrosophic Graphs with Real Life Applications." Mathematics 7, no. 6 (2019): 551. http://dx.doi.org/10.3390/math7060551.

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Fuzzy graph theory is a useful and well-known tool to model and solve many real-life optimization problems. Since real-life problems are often uncertain due to inconsistent and indeterminate information, it is very hard for an expert to model those problems using a fuzzy graph. A neutrosophic graph can deal with the uncertainty associated with the inconsistent and indeterminate information of any real-world problem, where fuzzy graphs may fail to reveal satisfactory results. The concepts of the regularity and degree of a node play a significant role in both the theory and application of graph

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Yakavi, AL, and Bibi. A. Mydeen. "Generalization of IP Domination Number for Complement Graphs." Indian Journal of Science and Technology 18, no. 12 (2025): 969–73. https://doi.org/10.17485/IJST/v18i12.3959.

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Abstract <strong>Objectives:</strong> This research introduces Isolate Pendant domination parameter, a novel concept applied to complement graphs. <strong>Methods:</strong> This investigation explores Isolate Pendant domination in complement graphs, characterizing the minimum dominating set and satisfying isolate pendant domination conditions within induced graphs, by applying this parameter in −𝐺 to yield insightful results. <strong>Findings:</strong> This study determines the minimum Isolate Pendant Dominating set (IPD-set) in complement graphs, characterizing

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Alsharafi, Mohammed, Abdu Alameri, Yusuf Zeren, Mahioub Shubatah, and Anwar Alwardi. "The Y-Index of Some Complement Graph Structures and Their Applications of Nanotubes and Nanotorus." Journal of Mathematics 2024 (January 17, 2024): 1–17. http://dx.doi.org/10.1155/2024/4269325.

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Topological descriptors play a significant role in chemical nanostructures. These topological measures have explicit chemical uses in chemistry, medicine, biology, and computer sciences. This study calculates the Y-index of some graphs and complements graph operations such as join, tensor and Cartesian and strong products, composition, disjunction, and symmetric difference between two simple graphs. Moreover, the Y-polynomial of titania nanotubes and the formulae for the Y-index, Y-polynomial, F-index, F-polynomial, and Y-coindex of the HAC5C7q,p and HAC5C6C7q,p nanotubes and their molecular c

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Lee, Eon-Kyung, and Sang-Jin Lee. "Embeddability of right-angled Artin groups on complements of trees." International Journal of Algebra and Computation 28, no. 03 (2018): 381–94. http://dx.doi.org/10.1142/s0218196718500182.

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For a finite simplicial graph [Formula: see text], let [Formula: see text] denote the right-angled Artin group on [Formula: see text]. Recently, Kim and Koberda introduced the extension graph [Formula: see text] for [Formula: see text], and established the Extension Graph Theorem: for finite simplicial graphs [Formula: see text] and [Formula: see text], if [Formula: see text] embeds into [Formula: see text] as an induced subgraph then [Formula: see text] embeds into [Formula: see text]. In this paper, we show that the converse of this theorem does not hold for the case [Formula: see text] is t

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41

Raksha, M. R., P. Hithavarshini, Charles Dominic, and N. K. Sudev. "Injective coloring of complementary prism and generalized complementary prism graphs." Discrete Mathematics, Algorithms and Applications 12, no. 02 (2020): 2050026. http://dx.doi.org/10.1142/s1793830920500263.

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The complementary prism [Formula: see text] of a graph [Formula: see text] is the graph obtained by drawing edges between the corresponding vertices of a graph [Formula: see text] and its complement [Formula: see text]. In this paper, we generalize the concept of complementary prisms of graphs and determine the injective chromatic number of generalized complementary prisms of graphs. We prove that for any simple graph [Formula: see text] of order [Formula: see text], [Formula: see text] and if [Formula: see text] is a graph with a universal vertex, then [Formula: see text].

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Hamidi, Mohammad, and Florentin Smarandache. "Valued-inverse Dombi neutrosophic graph and application." AIMS Mathematics 8, no. 11 (2023): 26614–31. http://dx.doi.org/10.3934/math.20231361.

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<abstract><p>Utilizing two ideas of neutrosophic subsets (NS) and triangular norms, we introduce a new type of graph as valued-inverse Dombi neutrosophic graphs. The valued-inverse Dombi neutrosophic graphs are a generalization of inverse neutrosophic graphs and are dual to Dombi neutrosophic graphs. We present the concepts of (complete) strong valued-inverse Dombi neutrosophic graphs and analyze the complement of (complete) strong valued-inverse Dombi neutrosophic graphs and self-valued complemented valued-inverse Dombi neutrosophic graphs. Since the valued-inverse Dombi neutrosop

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Wang, Haiying, Muhammad Javaid, Sana Akram, Muhammad Jamal, and Shaohui Wang. "Least eigenvalue of the connected graphs whose complements are cacti." Open Mathematics 17, no. 1 (2019): 1319–31. http://dx.doi.org/10.1515/math-2019-0097.

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Abstract Suppose that Γ is a graph of order n and A(Γ) = [ai,j] is its adjacency matrix such that ai,j is equal to 1 if vi is adjacent to vj and ai,j is zero otherwise, where 1 ≤ i, j ≤ n. In a family of graphs, a graph is called minimizing if the least eigenvalue of its adjacency matrix is minimum in the set of the least eigenvalues of all the graphs. Petrović et al. [On the least eigenvalue of cacti, Linear Algebra Appl., 2011, 435, 2357-2364] characterized a minimizing graph in the family of all cacti such that the complement of this minimizing graph is disconnected. In this paper, we chara

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Liu, Jia-Bao, Bahadur Ali, Muhammad Aslam Malik, Hafiz Muhammad Afzal Siddiqui, and Muhammad Imran. "Reformulated Zagreb Indices of Some Derived Graphs." Mathematics 7, no. 4 (2019): 366. http://dx.doi.org/10.3390/math7040366.

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A topological index is a numeric quantity that is closely related to the chemical constitution to establish the correlation of its chemical structure with chemical reactivity or physical properties. Miličević reformulated the original Zagreb indices in 2004, replacing vertex degrees by edge degrees. In this paper, we established the expressions for the reformulated Zagreb indices of some derived graphs such as a complement, line graph, subdivision graph, edge-semitotal graph, vertex-semitotal graph, total graph, and paraline graph of a graph.

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Subhash Mallinath Gaded and Nithya Sai Narayana. "On zero divisors graphs of direct product of finite fields." Journal of Computational Mathematica 7, no. 1 (2023): 100–109. http://dx.doi.org/10.26524/cm165.

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Zero divisor graphs are a fascinating area of current research. In this article, we look at some basic properties of zero divisor graphs of direct products of finite fields. We determine the girth, diameter, planarity, total domination number, connected domination number of the zero divisor graph as well as the complement graph of zero divisor graphs of direct products of finite fields.

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Das, Angsuman. "Coefficient of domination in graph." Discrete Mathematics, Algorithms and Applications 09, no. 02 (2017): 1750018. http://dx.doi.org/10.1142/s1793830917500185.

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This paper introduces a new domination parameter called coefficient of domination of a graph, which measures the maximum possible efficiency of domination for the given graph. It is evaluated for various classes of graphs and different bounds are proposed. Finally, Nordhaus–Gaddum type inequalities are established for coefficient of domination of a graph and its complement.

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R., Parimaleswari, Anbalagan S. та Subiramaniyan. "VERTEX COLORING OF (α-CUT) COMPLEMENT FUZZY GRAPH". International Journal of Current Research and Modern Education 3, № 1 (2018): 574–78. https://doi.org/10.5281/zenodo.1407321.

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Let G=( )  be a simple connected undirected fuzzy graph where  is a fuzzy set of vertices where each vertices has membership value µ and  is a fuzzy set of edges where each edge has a membership value . Vertex coloring is a function which assign colors to the vertices so that adjacent vertices receive different colors. We have examined the vertex coloring of complement fuzzy graph through the -cuts and found the chromatic number of that fuzzy graph.

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Ranasinghe, P. G. R. S., and L. R. M. K. R. Jayathilaka. "On the Pendant Number of Certain Graphs." Journal of Advances in Mathematics and Computer Science 38, no. 4 (2023): 33–41. http://dx.doi.org/10.9734/jamcs/2023/v38i41756.

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The present study investigates the pendant number of certain graph classes; complement, line graphs, and total graphs. The pendant number is the minimum number of end vertices of paths in a path decomposition of a graph. A path decomposition of a graph is a decomposition of it into subgraphs; i.e., a sequence of a subset of vertices of the graph such that the endpoints of each edge appear in one of the subsets and each vertex appears in an adjacent sub-sequence of the subsets.

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Rodríguez, José M., and José M. Sigarreta. "The hyperbolicity constant of infinite circulant graphs." Open Mathematics 15, no. 1 (2017): 800–814. http://dx.doi.org/10.1515/math-2017-0061.

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Abstract If X is a geodesic metric space and x1, x2, x3 ∈ X, a geodesic triangle T = {x1, x2, x3} is the union of the three geodesics [x1x2], [x2x3] and [x3x1] in X. The space X is δ-hyperbolic (in the Gromov sense) if any side of T is contained in a δ-neighborhood of the union of the two other sides, for every geodesic triangle T in X. Deciding whether or not a graph is hyperbolic is usually very difficult; therefore, it is interesting to find classes of graphs which are hyperbolic. A graph is circulant if it has a cyclic group of automorphisms that includes an automorphism taking any vertex

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Chandrasekar, K. Raja, and S. Saravanakumar. "OPEN PACKING NUMBER FOR SOME CLASSES OF PERFECT GRAPHS." Ural Mathematical Journal 6, no. 2 (2020): 38. http://dx.doi.org/10.15826/umj.2020.2.004.

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Let \(G\) be a graph with the vertex set \(V(G)\). A subset \(S\) of \(V(G)\) is an open packing set of \(G\) if every pair of vertices in \(S\) has no common neighbor in \(G.\) The maximum cardinality of an open packing set of \(G\) is the open packing number of \(G\) and it is denoted by \(\rho^o(G)\). In this paper, the exact values of the open packing numbers for some classes of perfect graphs, such as split graphs, \(\{P_4, C_4\}\)-free graphs, the complement of a bipartite graph, the trestled graph of a perfect graph are obtained.

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