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Journal articles on the topic 'Graceful graph'

Author: Grafiati

Published: 4 June 2021

Last updated: 13 February 2022

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1

Zeen El Deen, Mohamed R., and Nora A. Omar. "Extending of Edge Even Graceful Labeling of Graphs to Strong r -Edge Even Graceful Labeling." Journal of Mathematics 2021 (April 2, 2021): 1–19. http://dx.doi.org/10.1155/2021/6643173.

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Edge even graceful labeling of a graph G with p vertices and q edges is a bijective f from the set of edge E G to the set of positive integers 2,4 , … , 2 q such that all the vertex labels f ∗ V G , given by f ∗ u = ∑ u v ∈ E G f u v mod 2 k , where k = max p , q , are pairwise distinct. There are many graphs that do not have edge even graceful labeling, so in this paper, we have extended the definition of edge even graceful labeling to r -edge even graceful labeling and strong r -edge even graceful labeling. We have obtained the necessary conditions for more path-related graphs and cycle-related graphs to be an r -edge even graceful graph. Furthermore, the minimum number r for which the graphs: tortoise graph, double star graph, ladder and diagonal ladder graphs, helm graph, crown graph, sunflower graph, and sunflower planar graph, have an r -edge even graceful labeling was found. Finally, we proved that the even cycle C 2 n has a strong 2 -edge even graceful labeling when n is even.

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2

Eshghi, Kourosh, and Parham Azimi. "Applications of mathematical programming in graceful labeling of graphs." Journal of Applied Mathematics 2004, no. 1 (2004): 1–8. http://dx.doi.org/10.1155/s1110757x04310065.

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Graceful labeling is one of the best known labeling methods of graphs. Despite the large number of papers published on the subject of graph labeling, there are few particular techniques to be used by researchers to gracefully label graphs. In this paper, first a new approach based on the mathematical programming technique is presented to model the graceful labeling problem. Then a branching method is developed to solve the problem for special classes of graphs. Computational results show the efficiency of the proposed algorithm for different classes of graphs. One of the interesting results of our model is in the class of trees. The largest tree known to be graceful has at most 27 vertices but our model can easily solve the graceful labeling for trees with 40 vertices.

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3

V J Kaneria, H P Chudasama, and P P Andharia. "Absolute Mean Graceful Labeling in Path Union of Various Graphs." Mathematical Journal of Interdisciplinary Sciences 7, no. 1 (September 6, 2018): 51–56. http://dx.doi.org/10.15415/mjis.2018.71008.

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Present paper aims to focus on absolute mean graceful labeling in path union of various graphs. We proved path union of graphs like tree, path Pn, cycle Cn, complete bipartite graph Km, n, grid graph PM × Pn, step grid graph Stn and double step grid graph DStn are absolute mean graceful graphs.

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4

Semenyuta, Marina F. "FIBONACCI AND SUPER FIBONACCI GRACEFUL LABELLINGS OF SOME TYPES OF GRAPHS." Journal of Automation and Information sciences 1 (January 1, 2021): 105–21. http://dx.doi.org/10.34229/0572-2691-2021-1-10.

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We consider the basic theoretical information regarding the Fibonacci graceful graphs. An injective function is said a Fibonacci graceful labelling of a graph of a size , if it induces a bijective function on the set of edges , where by the rule , for any adjacent vertices A graph that allows such labelling is called Fibonacci graceful. In this paper, we introduce the concept of super Fibonacci graceful labelling, narrowing the set of vertex labels, i.e. Four types of problems to be studied are selected. In the problem of the first type, the following question is raised: is there a graph that allows a certain kind of labelling, and under what conditions does this take place? The problem of the second type is the problem of construction: it is necessary, for a given system of requirements for the graph, to construct (at least one) its labelling that would satisfy this system. The following two types of problems relate to enumeration problems: for a given graph, determine the number of different Fibonacci and / or super Fibonacci graceful labellings; build all the different labellings of a given kind. As a result of solving these problems, functions were found that generate Fibonacci and super Fibonacci graceful labellings for graphs of cyclic structure; necessary and sufficient conditions for the existence of Fibonacci graceful labelling for disjunctive union of cycles, super Fibonacci graceful labelling for cycles, Eulerian graphs are obtained; the number of non-equivalent labellings of the cycle is determined; conditions for the existence of a super Fibonacci graceful labelling of a one-point connection of arbitrary connected super Fibonacci graceful graphs … …, are presented

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5

Beatress, N. Adalin, and P. B. SARASIJA P.B. SARASIJA. "An Arithmetic Edge Graceful Graph." International Journal of Scientific Research 2, no. 11 (June 1, 2012): 345–46. http://dx.doi.org/10.15373/22778179/nov2013/109.

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6

Bhutani, Kiran R., and Alexander B. Levin. "Graceful numbers." International Journal of Mathematics and Mathematical Sciences 29, no. 8 (2002): 495–99. http://dx.doi.org/10.1155/s0161171202007615.

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We construct a labeled graphD(n)that reflects the structure of divisors of a given natural numbern. We define the concept of graceful numbers in terms of this associated graph and find the general form of such a number. As a consequence, we determine which graceful numbers are perfect.

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7

Kaneria, V. J., H. M. Makadia, and R. V. Viradia. "Graceful Labeling for Disconnected Grid Related Graphs." Bulletin of Mathematical Sciences and Applications 11 (February 2015): 6–11. http://dx.doi.org/10.18052/www.scipress.com/bmsa.11.6.

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In this paper we have proved that union of three grid graphs, U3l=1(Pnl×Pml)and union of finite copies of a grid graph (Pn×Pm)are graceful. We have also given two graceful labeling functions to the grid graph (Pn×Pm).

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8

Pasaribu, Meliana, Yundari Yundari, and Muhammad Ilyas. "Graceful Labeling and Skolem Graceful Labeling on the U-star Graph and It’s Application in Cryptography." Jambura Journal of Mathematics 3, no. 2 (May 4, 2021): 103–14. http://dx.doi.org/10.34312/jjom.v3i2.9992.

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Graceful Labeling on graph G=(V, E) is an injective function f from the set of the vertex V(G) to the set of numbers {0,1,2,...,|E(G)|} which induces bijective function f from the set of edges E(G) to the set of numbers {1,2,...,|E(G)|} such that for each edge uv e E(G) with u,v e V(G) in effect f(uv)=|f(u)-f(v)|. Meanwhile, the Skolem graceful labeling is a modification of the Graceful labeling. The graph has graceful labeling or Skolem graceful labeling is called graceful graph or Skolem graceful labeling graph. The graph used in this study is the U-star graph, which is denoted by U(Sn). The purpose of this research is to determine the pattern of the graceful labeling and Skolem graceful labeling on graph U(Sn) apply it to cryptography polyalphabetic cipher. The research begins by forming a graph U(Sn) and they are labeling it with graceful labeling and Skolem graceful labeling. Then, the labeling results are applied to the cryptographic polyalphabetic cipher. In this study, it is found that the U(Sn) graph is a graceful graph and a Skolem graceful graph, and the labeling pattern is obtained. Besides, the labeling results on a graph it U(Sn) can be used to form a table U(Sn) polyalphabetic cipher. The table is used as a key to encrypt messages.

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9

Meng Lin, Yanyou Chai, Jinhao Jiang, and Guojing Wang. "Some Conclusions about Graceful Graph and k-graceful Graph Properties." International Journal of Advancements in Computing Technology 5, no. 2 (January 31, 2013): 484–93. http://dx.doi.org/10.4156/ijact.vol5.issue2.61.

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10

Hosseini, Sayed Mohammad, Mahdi Davoudi Darareh, Shahrooz Janbaz, and Ali Zaghian. "An Adiabatic Quantum Algorithm for Determining Gracefulness of a Graph." Zeitschrift für Naturforschung A 72, no. 7 (July 26, 2017): 637–45. http://dx.doi.org/10.1515/zna-2017-0011.

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AbstractGraph labelling is one of the noticed contexts in combinatorics and graph theory. Graceful labelling for a graph G with e edges, is to label the vertices of G with 0, 1, ℒ, e such that, if we specify to each edge the difference value between its two ends, then any of 1, 2, ℒ, e appears exactly once as an edge label. For a given graph, there are still few efficient classical algorithms that determine either it is graceful or not, even for trees – as a well-known class of graphs. In this paper, we introduce an adiabatic quantum algorithm, which for a graceful graph G finds a graceful labelling. Also, this algorithm can determine if G is not graceful. Numerical simulations of the algorithm reveal that its time complexity has a polynomial behaviour with the problem size up to the range of 15 qubits. A general sufficient condition for a combinatorial optimization problem to have a satisfying adiabatic solution is also derived.

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Bantara, J. A., I. W. Sudarsana, and S. Musdalifah. "PELABELAN GRACEFUL GANJIL PADA GRAF DUPLIKASI DAN SPLIT BINTANG." JURNAL ILMIAH MATEMATIKA DAN TERAPAN 15, no. 1 (May 14, 2018): 28–35. http://dx.doi.org/10.22487/2540766x.2018.v15.i1.10193.

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Graph is not an empty a finite of the objects that called point (vertex) with the couple was not that is the side (edge). The set point denoted by , while the set edge denoted by . Odd graceful labeling on graph with side is a function injective from so that induced function such that in label with so label sides would be different. A graph that have an odd graceful labeling is called odd graceful graph. The result showed that duplicate star graph for and split star graph for , for even satisfie odd graceful labeling.

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A. Mariadoss, Swaminathan, and Sunita D';Silva. "Diophantine Edge Graceful Graph." Recent Patents on Computer Science 9, no. 3 (January 16, 2017): 190–94. http://dx.doi.org/10.2174/2213275908666151028192622.

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13

Manulang, Jona Martinus, and Kiki A. Sugeng. "Graceful labeling on torch graph." Indonesian Journal of Combinatorics 2, no. 1 (June 12, 2018): 14. http://dx.doi.org/10.19184/ijc.2018.2.1.2.

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Let G be a graph with vertex set V=V(G) and edge set E=E(G). An injective function f:V<span style="font-family: symbol;"> --> </span>{0,1,2,...,|E|} is called graceful labeling if f induces a function f<sup>*</sup>(uv)=|f(u)<span style="font-family: symbol;">-</span>f(v)| which is a bijection from E(G) to the set {1,2,3,...,|E|}. A graph which admits a graceful labeling is called a graceful graph. In this paper, we show that torch graph O<sub>n</sub> is a graceful graph.

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14

Taweel, Faris M., Eman A. Abuhijleh, and Shorouq Ali. "A Graceful Labeling of Square of Path Graph with Quadratic Complexity Algorithm." WSEAS TRANSACTIONS ON MATHEMATICS 20 (August 23, 2021): 378–86. http://dx.doi.org/10.37394/23206.2021.20.39.

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A method for relaxed graceful labeling of P2n graphs is presented together with an algorithm designed for labeling these graphs. Graceful labeling is achieved by relaxing the range to 2m and perform the labeling using an algorithm with quadratic complexity (O(n2)). The algorithm can be used for labeling any P2n graph with n ≥ 3, as far as the machine can handle the size of the problem.

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15

GAO, ZHEN-BIN, XIAO-DONG ZHANG, and LI-JUAN XU. "ODD GRACEFUL LABELINGS OF GRAPHS." Discrete Mathematics, Algorithms and Applications 01, no. 03 (September 2009): 377–88. http://dx.doi.org/10.1142/s1793830909000300.

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A graph G = (V(G), E(G)) with q edges is said to be odd graceful if there exists an injection f from V(G) to {0, 1, 2, …, 2q - 1} such that the edge labeling set is {1, 3, 5, …, 2q - 1} with each edge xy assigned the label |f(x) - f(y)|. In this paper, we prove that Pn × Pm (m = 2, 3, 4), generalized crown graphs Cn ⊙ K1,t and gear graphs are odd graceful.

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16

Kaneria, V. J., H. M. Makadia, and R. V. Viradia. "Various Graph Operations on Semi Smooth Graceful Graphs." International Journal of Mathematics and Soft Computing 6, no. 1 (January 9, 2016): 57. http://dx.doi.org/10.26708/ijmsc.2016.1.6.06.

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17

Zeen El Deen, Mohamed R. "Edge δ− Graceful Labeling for Some Cyclic-Related Graphs." Advances in Mathematical Physics 2020 (January 20, 2020): 1–18. http://dx.doi.org/10.1155/2020/6273245.

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In this paper, we introduce a new type of labeling of a graph G with p vertices and q edges called edge δ− graceful labeling, for any positive integer δ, as a bijective mapping f of the edge set EG into the set δ,2δ,3δ,⋯,qδ such that the induced mapping f∗:VG→0,δ,2δ,3δ,⋯,qδ−δ, given by f∗u=∑uv∈EGfuvmodδk, where k=maxp,q, is an injective function. We prove the existence of an edge δ− graceful labeling, for any positive integer δ, for some cycle-related graphs like the wheel graph, alternate triangular cycle, double wheel graph Wn,n, the prism graph Πn, the prism of the wheel PWn, the gear graph Gn, the closed helm CHn, the butterfly graph Bn, and the friendship Frn.

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18

Anitha, K., B. Selvam, and K. Thirusangu. "K-Graceful, Odd-Even Graceful, Heronian Mean and Analytic Mean Labeling for the Extended Duplicate Graph of Kite Graph." International Journal of Engineering & Technology 7, no. 4.10 (October 2, 2018): 355. http://dx.doi.org/10.14419/ijet.v7i4.10.20934.

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19

Daoud, Salama Nagy. "Edge Even Graceful Labeling of Polar Grid Graphs." Symmetry 11, no. 1 (January 2, 2019): 38. http://dx.doi.org/10.3390/sym11010038.

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Edge Even Graceful Labelingwas first defined byElsonbaty and Daoud in 2017. An edge even graceful labeling of a simple graph G with p vertices and q edges is a bijection f from the edges of the graph to the set { 2 , 4 , … , 2 q } such that, when each vertex is assigned the sum of all edges incident to it mod 2 r where r = max { p , q } , the resulting vertex labels are distinct. In this paper we proved necessary and sufficient conditions for the polar grid graph to be edge even graceful graph.

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20

Mahmoudzadeh, Houra, and Kourosh Eshghi. "A Metaheuristic Approach to the Graceful Labeling Problem." International Journal of Applied Metaheuristic Computing 1, no. 4 (October 2010): 42–56. http://dx.doi.org/10.4018/jamc.2010100103.

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In graph theory, a graceful labeling of a graph G = (V, E) with n vertices and m edges is a labeling of its vertices with distinct integers between 0 and m inclusive, such that each edge is uniquely identified by the absolute difference between its endpoints. In this paper, the well-known graceful labeling problem of graphs is represented as an optimization problem, and an algorithm based on Ant Colony Optimization metaheuristic is proposed for finding its solutions. In this regard, the proposed algorithm is applied to different classes of graphs and the results are compared with the few existing methods inside of different literature.

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21

Hameed Hassan, T., and R. Mohammad Abbas. "Prime Graceful Labeling." International Journal of Engineering & Technology 7, no. 4.36 (December 9, 2018): 750. http://dx.doi.org/10.14419/ijet.v7i4.36.24234.

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A graph G with m vertices and n edges, is said to be prime graceful labeling, if there is an injection from the vertices of G to {1, 2, ..., k} where k = min {2m, 2n} such that gcd ( ( ), ( )=1 and the induced injective function from the edges of G to {1, 2, ..., k − 1} defined by ( ) = | ( ) − ( ) | , the resulting edge labels are distinct. In this paper path , cycle Cn , star K1,n , friendship graph Fn , bistar Bn,n, C4 ∪ Pn , Km,2 and Km,2 ∪ Pn are shown to be Prime Graceful Labeling .

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Senthil Kumar, D. "SUPER FIBONACCI GRACEFUL LABELING FOR GENERALIZED (a,m)-SHELL GRAPH AND MERGED WITH SOME GRAPHS." Advances in Mathematics: Scientific Journal 9, no. 11 (November 8, 2020): 9813–17. http://dx.doi.org/10.37418/amsj.9.11.91.

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A graph $G$ with $p$ vertices and $q$ edges has super Fibonacci graceful labeling if there exists an injective map $f : V(G) \rightarrow \left\{F_{0}, F_{1},F_{2},\ldots F_{q}\right\}$ where $F_{k}$ is the $k^{th}$ Fibonacci number of the Fibonacci series such that its induced map $f^{+}: E(G) \rightarrow\left\{F_{1},F_{2},F_{3},\ldots F_{q}\right\}$ defined by $f^{+}(xy)$ =$\left|f(x) - f(y)\right|$ $\forall$ $xy \in G,$ is a bijective map. In this paper, we investigate the existence of super Fibonacci graceful labeling for the various types of $(a, m)$ - shell graph and $(a, m)$ - shell graph merged with some graphs.

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23

El-Zohny, H., S. Radwan, S. I. Abo El-Fotooh, and Z. Mohammed. "Graceful Labeling of Hypertrees." Journal of Mathematics Research 13, no. 1 (January 11, 2021): 28. http://dx.doi.org/10.5539/jmr.v13n1p28.

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Graph labeling is considered as one of the most interesting areas in graph theory. A labeling for a simple graph G (numbering or valuation), is an association of non -negative integers to vertices of G  (vertex labeling) or to edges of G  (edge labeling) or both of them. In this paper we study the graceful labeling for the k- uniform hypertree and define a condition for the corresponding tree to be graceful. A k- uniform hypertree is graceful if the minimum difference of vertices’ labels of each edge is distinct and each one is the label of the corresponding edge.

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El-Zohny, H., S. Radwan, S. I. Abo El-Fotooh, and Z. Mohammed. "Graceful Labeling of Hypertrees." Journal of Mathematics Research 13, no. 1 (January 11, 2021): 28. http://dx.doi.org/10.5539/jmr.v13n1p28.

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Graph labeling is considered as one of the most interesting areas in graph theory. A labeling for a simple graph G (numbering or valuation), is an association of non -negative integers to vertices of G  (vertex labeling) or to edges of G  (edge labeling) or both of them. In this paper we study the graceful labeling for the k- uniform hypertree and define a condition for the corresponding tree to be graceful. A k- uniform hypertree is graceful if the minimum difference of vertices’ labels of each edge is distinct and each one is the label of the corresponding edge.

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25

Panpa, A., and T. Poomsa-ard. "On Graceful Spider Graphs with at Most Four Legs of Lengths Greater than One." Journal of Applied Mathematics 2016 (2016): 1–5. http://dx.doi.org/10.1155/2016/5327026.

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A graceful labeling of a tree T with n edges is a bijection f:V(T)→{0,1,2,…,n} such that {|f(u)-f(v)|:uv∈E(T)} equal to {1,2,…,n}. A spider graph is a tree with at most one vertex of degree greater than 2. We show that all spider graphs with at most four legs of lengths greater than one admit graceful labeling.

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26

Liang, Zhihe, and Huijuan Zuo. "On the gracefulness of the graph P 2m,2n." Applicable Analysis and Discrete Mathematics 4, no. 1 (2010): 175–80. http://dx.doi.org/10.2298/aadm1000003l.

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Let Pa,b denotes a graph obtained by identifying the end vertices of b internally disjoint paths each of length a. Kathiresan conjectured that graph Pa,b is graceful except when a is odd and b ? 2 (mod 4). In this paper we show that the graph Pa,b is graceful when both a and b are even.

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Elsonbaty, Ahmed A., and Salama Nagy Daoud. "Edge Even Graceful Labeling of Cylinder Grid Graph." Symmetry 11, no. 4 (April 22, 2019): 584. http://dx.doi.org/10.3390/sym11040584.

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Edge even graceful labeling (e.e.g., l.) of graphs is a modular technique of edge labeling of graphs, introduced in 2017. An e.e.g., l. of simple finite undirected graph G = ( V ( G ) , E ( G ) ) of order P = | ( V ( G ) | and size q = | E ( G ) | is a bijection f : E ( G ) → { 2 , 4 , … , 2 q } , such that when each vertex v ∈ V ( G ) is assigned the modular sum of the labels (images of f ) of the edges incident to v , the resulting vertex labels are distinct mod 2 r , where r = max ( p , q ) . In this work, the family of cylinder grid graphs are studied. Explicit formulas of e.e.g., l. for all of the cases of each member of this family have been proven.

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28

Burzio, M., and G. Ferrarese. "The subdivision graph of a graceful tree is a graceful tree." Discrete Mathematics 181, no. 1-3 (February 1998): 275–81. http://dx.doi.org/10.1016/s0012-365x(97)00069-1.

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29

Ulaganathan, P. P. "Graceful and Skolem graceful Labeling in Extended Duplicate Graph of Path." Indian Journal of Science and Technology 4, no. 2 (February 20, 2011): 107–11. http://dx.doi.org/10.17485/ijst/2011/v4i2.5.

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Venkatesh, S., and K. Balasubramanian. "Some Results on Generating Graceful Trees." International Journal of Engineering & Technology 7, no. 4.10 (October 2, 2018): 570. http://dx.doi.org/10.14419/ijet.v7i4.10.21283.

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Let and be any two simple graphs. Then is the graph obtained by merging a vertex of each copy of with every attachment vertices of . Let be the one vertex union of copies of the given caterpillar with the common vertex as one of the penultimate vertices. If is any caterpillar, then define . Recursively for , construct ,that is, Here the tree considered for attachment with is a caterpillar, but not necessarily the same among the levels. In this paper we prove that the tree is graceful for

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31

LEE, SIN-MIN, and KAM-CHUEN NG. "Every Young Tableau Graph Is d-Graceful." Annals of the New York Academy of Sciences 555, no. 1 Combinatorial (May 1989): 296–302. http://dx.doi.org/10.1111/j.1749-6632.1989.tb22463.x.

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Raju, V., and M. Paruvatha vathana. "Graceful Labeling for Some Complete Bipartite Graph." Journal of Computer and Mathematical Sciences 9, no. 12 (December 12, 2018): 2147–52. http://dx.doi.org/10.29055/jcms/960.

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Sujatha, N., C. Dharuman, and K. Thirusangu. "Triangular fuzzy graceful labeling on star graph." Journal of Physics: Conference Series 1377 (November 2019): 012023. http://dx.doi.org/10.1088/1742-6596/1377/1/012023.

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Wang, Tao-Ming, Cheng-Chang Yang, Lih-Hsing Hsu, and Eddie Cheng. "Infinitely many equivalent versions of the graceful tree conjecture." Applicable Analysis and Discrete Mathematics 9, no. 1 (2015): 1–12. http://dx.doi.org/10.2298/aadm141009017w.

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A graceful labeling of a graph with q edges is a labeling of its vertices using the integers in [0, q], such that no two vertices are assigned the same label and each edge is uniquely identified by the absolute difference between the labels of its endpoints. The well known Graceful Tree Conjecture (GTC) states that all trees are graceful, and it remains open. It was proved in 1999 by Broersma and Hoede that there is an equivalent conjecture for GTC stating that all trees containing a perfect matching are strongly graceful (graceful with an extra condition). In this paper we extend the above result by showing that there exist infinitely many equivalent versions of the GTC. Moreover we verify these infinitely many equivalent conjectures of GTC for trees of diameter at most 7. Among others we are also able to identify new graceful trees and in particular generalize the ?-construction of Stanton-Zarnke (and later Koh- Rogers-Tan) for building graceful trees through two smaller given graceful trees.

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Ambika, B., and G. Balasubramanian. "EDGE ODD GRACEFUL LABELING OF SOME FLOWER PETAL GRAPHS." Advances in Mathematics: Scientific Journal 9, no. 11 (November 6, 2020): 9523–25. http://dx.doi.org/10.37418/amsj.9.11.54.

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A labeling of a graph $G$ with $\alpha$ vertices and $\beta$ edges called an edge odd graceful labeling if there is an edge labeling with odd numbers to all edges such that each vertex is assigned a label which is the sum $\mod (2\gamma)$ of labels of edge incident on it, where $\gamma=max\{\alpha,\beta\}$ and the induced vertex labels are distinct. In this paper, we discussed about edge odd gracefulness of some special class of flower petal graphs.

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Daoud, Salama Nagy, and Wedad Saleh. "Edge even graceful labeling of torus grid graph." Proyecciones (Antofagasta) 39, no. 4 (August 1, 2020): 1033–82. http://dx.doi.org/10.22199/issn.0717-6279-2020-04-0065.

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Nurvazly, D. E., and K. A. Sugeng. "Graceful Labelling of Edge Amalgamation of Cycle Graph." Journal of Physics: Conference Series 1108 (November 2018): 012047. http://dx.doi.org/10.1088/1742-6596/1108/1/012047.

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Mitra, Sarbari, and Soumya Bhoumik. "Graceful labeling of triangular extension of complete bipartite graph." Electronic Journal of Graph Theory and Applications 7, no. 1 (April 10, 2019): 11–30. http://dx.doi.org/10.5614/ejgta.2019.7.1.2.

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Jesintha, J. Jeba, K. Subashini, and Allu Merin Sabu. "Graceful labeling on twig diamond graph with pendant edges*." Bulletin of Pure & Applied Sciences- Mathematics and Statistics 39e, no. 2 (2020): 188–92. http://dx.doi.org/10.5958/2320-3226.2020.00018.1.

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Rajeswari, V., and K. Thiagarajan. "Graceful Labeling of Wheel Graph and Middle Graph of Wheel Graph under IBEDE and SIBEDE Approach." Journal of Physics: Conference Series 1000 (April 2018): 012078. http://dx.doi.org/10.1088/1742-6596/1000/1/012078.

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FUJITA, SATOSHI, AKIRA OHTSUBO, and MASAYA MITO. "EXTENDED SKIP GRAPHS FOR EFFICIENT KEY SEARCH IN PEER-TO-PEER ENVIRONMENT." Journal of Interconnection Networks 08, no. 02 (June 2007): 119–32. http://dx.doi.org/10.1142/s021926590700193x.

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In this paper, we propose three techniques to improve the cost/performance of the skip graph that was recently proposed by Aspnes and Shah. The skip graph, which is a distributed data structure that could efficiently support find, insert, and delete operations of a key drawn from a totally ordered set, consists of N nodes each of which is connected with exactly log 2 N nodes determined by a set of random binary vectors called membership vectors. In the following, we will extend the construction of the skip graph in the following two directions: 1) proposal of a subgraph of the skip graph which realizes a graceful degradation of the routing performance when the number of neighbors reduces from log 2 N, and 2) proposal of a supergraph of the skip graph which realizes a significant performance improvement when the number of neighbors increases from log 2 N. The performance of those extended graphs will be evaluated analytically.

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42

Hussain, R. Jahir, and R. M. Karthik Keyan. "The Kp - Bondage And Kp - Non Bondage Number Of Fuzzy Graphs And Graceful Graph." IOSR Journal of Electrical and Electronics Engineering 12, no. 03 (July 2017): 10–20. http://dx.doi.org/10.9790/1676-1203051020.

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43

H. P, Chudasama, and Kaneria V. J. "Duplication of some graph elements and absolute mean graceful labeling." International Journal of Mathematics Trends and Technology 65, no. 4 (April 25, 2019): 107–15. http://dx.doi.org/10.14445/22315373/ijmtt-v65i4p520.

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Bhoumik, Soumya, and Sarbari Mitra. "Graceful labeling of pendant edge extension of complete bipartite graph." International Journal of Mathematical Analysis 8 (2014): 2885–97. http://dx.doi.org/10.12988/ijma.2014.410334.

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Khoirunnisa, S., Dafik, A. I. Kristiana, R. Alfarisi, and E. R. Albirri. "On graceful chromatic number of comb product of ladder graph." Journal of Physics: Conference Series 1836, no. 1 (March 1, 2021): 012027. http://dx.doi.org/10.1088/1742-6596/1836/1/012027.

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Kanani, J. C., and V. J. Kaneria. "Graceful labeling in a graph consisting chord with quadrilateral snake." Malaya Journal of Matematik 9, no. 1 (2021): 415–18. http://dx.doi.org/10.26637/mjm0901/0070.

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Xie, Jian Min, Bing Yao, Ming Yao, and Xiang En Chen. "Research on Information Process with a Computational Approach to Some Odd-Graceful Trees." Advanced Materials Research 1022 (August 2014): 207–10. http://dx.doi.org/10.4028/www.scientific.net/amr.1022.207.

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Graph labeling theory has important applications in coding theory, communication networks, logistics and other aspects. In Operations Research or Systems Engineering Theory and Methods, one very often use graph colorings/labellings to divide large systems into subsystems. One can use colorings/labellings to distinguish vertices and edges between vertices in order to find fast algorithms to imitate some effective transmissions and communications in information networks. In this paper we present a computational approach to the odd-graceful labelings for some olive trees.

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Solairaju, A., and P. Muruganantham. "Even Vertex Graceful of Path, Circuit, Star, Wheel, some Extension-friendship Graphs and Helm Graph." International Journal of Computer Applications 10, no. 6 (November 10, 2010): 5–8. http://dx.doi.org/10.5120/1488-2005.

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Sasikala, A., and C. Vimala. "EDGE ODD GRACEFUL LABELING OF UMBRELLA GRAPH AND N(2C3 + P2)." Advances in Mathematics: Scientific Journal 9, no. 3 (July 3, 2020): 1307–14. http://dx.doi.org/10.37418/amsj.9.3.76.

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Amri, Z., I. Irvan, I. Maryanti, and H. Sumardi. "Odd graceful labeling on the Ilalang graph (Sn , 3) it’s variation." Journal of Physics: Conference Series 1731 (January 2021): 012030. http://dx.doi.org/10.1088/1742-6596/1731/1/012030.

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