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Journal articles on the topic 'Graceful tree'

Author: Grafiati

Published: 3 June 2025

Last updated: 31 July 2025

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1

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

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2

Gobind, Mohanty, Mishra Debdas, Sarangi Pravat, and Bhattacharjee Subarna. "Some New Classes of (k, d) Graceful 3 Distance Trees and 3 Distance Unicyclic Graphs." Indian Journal of Science and Technology 15, no. 14 (2022): 630–39. https://doi.org/10.17485/IJST/v15i14.254.

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Abstract <strong>Objectives:</strong> To identify a new family of (k; d) graceful graphs. <strong>Methods :</strong> The methodology involves mathematical formulation for labeling of the vertices of a given graph and subsequently establishing that these formulations give rise to (k;d) graceful labeling. <strong>Findings:</strong> Here we define a three-distance tree as the tree possessing a path such that each vertex of the tree is at most at a distance three from that path. In this paper we identify two families of three distance trees that possess (k; d) graceful lab

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3

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 re

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4

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

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5

Kirubaharan, D. R., and Dr G. Nirmala. "Graceful V* 2Fn-tree." IOSR Journal of Mathematics 10, no. 2 (2014): 01–06. http://dx.doi.org/10.9790/5728-10240106.

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Abdullah Zahraa O, Arif Nabeel E, and F. A. Fawzi. "Dividing Graceful Labeling of Certain Tree Graphs." Tikrit Journal of Pure Science 25, no. 4 (2020): 123–26. http://dx.doi.org/10.25130/tjps.v25i4.281.

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A tree is a connected acyclic graph on n vertices and m edges. graceful labeling of a tree defined as a simple undirected graph G(V,E) with order n and size m, if there exist an injective mapping that induces a bijective mapping defined by for each and . In this paper we introduce a new type of graceful labeling denoted dividing graceful then discuss this type of certain tree graphs .

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7

Kristiana, Arika Indah, Ahmad Aji, Edy Wihardjo, and Deddy Setiawan. "on Graceful Chromatic Number of Vertex amalgamation of Tree Graph Family." CAUCHY: Jurnal Matematika Murni dan Aplikasi 7, no. 3 (2022): 432–44. http://dx.doi.org/10.18860/ca.v7i3.16334.

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Proper vertex coloring c of a graph G is a graceful coloring if c is a graceful k-coloring for k∈{1,2,3,…}. Definition graceful k-coloring of a graph G=(V,E) is a proper vertex coloring c:V(G)→{1,2,…,k);k≥2, which induces a proper edge coloring c':E(G)→{1,2,…,k-1} defined c'(uv)=|c(u)-c(v)|. The minimum vertex coloring from graph G can be colored with graceful coloring called a graceful chromatic number with notation χg (G). In this paper, we will investigate the graceful chromatic number of vertex amalgamation of tree graph family with some graph is path graph, centipede graph, broom and E gr

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8

V.Ramachandran and C.Sekar. "ONE MODULO N GRACEFULNESS OF REGULAR BAMBOO TREE AND COCONUT TREE." International journal on applications of graph theory in wireless ad hoc networks and sensor networks (GRAPH-HOC) 6, no. 2 (2014): 1–10. https://doi.org/10.5281/zenodo.3532228.

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A function f is called a graceful labelling of a graph G with q edges if f is an injection from the vertices of G to the set {0, 1, 2, . . . , q} such that, when each edge xy is assigned the label |f(x) − f(y)| , the resulting edge labels are distinct. A graph G is said to be one modulo N graceful (where N is a positive integer) if there is a function φ from the vertex set of G to {0, 1,N, (N + 1), 2N, (2N + 1), . . . ,N(q − 1),N(q − 1) + 1} in such a way that (i) φ is 1 − 1 (ii) φ induces a bijection φ_ from the edge set of G to {1,N + 1, 2N + 1, . . .

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9

Su, Jing, Hongyu Wang, and Bing Yao. "Matching-Type Image-Labelings of Trees." Mathematics 9, no. 12 (2021): 1393. http://dx.doi.org/10.3390/math9121393.

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A variety of labelings on trees have emerged in order to attack the Graceful Tree Conjecture, but lack showing the connections between two labelings. In this paper, we propose two new labelings: vertex image-labeling and edge image-labeling, and combine new labelings to form matching-type image-labeling with multiple restrictions. The research starts from the set-ordered graceful labeling of the trees, and we give several generation methods and relationships for well-known labelings and two new labelings on trees.

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10

Sethuraman, G., P. Ragukumar, and Peter J. Slater. "Embedding an Arbitrary Tree in a Graceful Tree." Bulletin of the Malaysian Mathematical Sciences Society 39, S1 (2015): 341–60. http://dx.doi.org/10.1007/s40840-015-0210-5.

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11

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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12

Saengsura, Kittisak, and Tiang Poomsa-ard. "Graceful Labeling of Some Spider Graphs." European Journal of Pure and Applied Mathematics 18, no. 2 (2025): 5305. https://doi.org/10.29020/nybg.ejpam.v18i2.5305.

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A graceful labeling of a tree $T$ with $n$ edges is a bijection $f : V(T) \longrightarrow \{0,1,2, \ldots n\}$ such that $\{|f(u)-f(v)| : uv \in E(T)\}$ equal to $\{1,2,3,\ldots,n\}$. A spider graph is a tree with one vertex of degree at least $3$ and all others with degree at most $2$. We show that some classes of spider graphs admit graceful labeling.

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Parvathi, N., and S. Vidyanandini. "Graceful Labeling of a Tree from Caterpillars." Journal of Information and Optimization Sciences 35, no. 4 (2014): 387–93. http://dx.doi.org/10.1080/02522667.2014.961811.

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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 (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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Afifah, Lilla, and I. Ketut Budayasa. "PELABELAN ANGGUN GRAF BERLIAN RANGKAP BERBINTANG, BEBERAPA KELAS GRAF POHON, DAN GRAF CORONA KHUSUS." MATHunesa: Jurnal Ilmiah Matematika 11, no. 3 (2023): 368–82. http://dx.doi.org/10.26740/mathunesa.v11n3.p368-382.

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Pelabelan dari suatu graf adalah suatu pemetaan yang membawa setiap elemen graf yaitu himpunan sisi (edge) atau himpunan titik (vertex) ke bilangan bilangan bulat positif, yang disebut label. Sebuah fungsi disebut pelabelan anggun graf dengan m sisi jika adalah injektif dan fungsi terinduksi didefinisikan sebagai adalah bijektif. Graf yang mempunyai pelabelan anggun disebut graf anggun. Pada penelitian ini akan ditunjukkan konstruksi pelabelan anggun pada graf berlian rangkap berbintang , beberapa kelas graf pohon dan graf corona khusus (K_(n,n) ⨀ K_1). Kata kunci: Pelabelan anggun, graf

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Venkatesh, S., and K. Balasubramanian. "Some Results on Generating Graceful Trees." International Journal of Engineering & Technology 7, no. 4.10 (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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17

Su, Jing, Hui Sun, and Bing Yao. "Odd-Graceful Total Colorings for Constructing Graphic Lattice." Mathematics 10, no. 1 (2021): 109. http://dx.doi.org/10.3390/math10010109.

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The security of passwords generated by the graphic lattices is based on the difficulty of the graph isomorphism, graceful tree conjecture, and total coloring conjecture. A graphic lattice is generated by a graphic base and graphical operations, where a graphic base is a group of disjointed, connected graphs holding linearly independent properties. We study the existence of graphic bases with odd-graceful total colorings and show graphic lattices by vertex-overlapping and edge-joining operations; we prove that these graphic lattices are closed to the odd-graceful total coloring.

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18

Mathew, Varkey T.K, and Kumar T.J Rajesh. "EVEN GRACEFUL LABELLING OF A CLASS OF TREES." International Journal on Applications of Graph Theory in Wireless Ad hoc Networks and Sensor Networks (GRAPH-HOC) 7, no. 4 (2020): 1–7. https://doi.org/10.5281/zenodo.3923752.

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A labelling or numbering of a graph G with q edges is an assignment of labels to the vertices of G that induces for each edge uv a labelling depending on the vertex labels f(u) and f(v). A labelling is called a graceful labelling if there exists an injective function f: V (G) → {0, 1,2,......q} such that for each edge xy, the labelling │f(x)-f(y)│is distinct. In this paper, we prove that a class of Tn trees are even graceful.

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Mathew, Varkey T.K, and Kumar T.J Rajesh. "EVEN GRACEFUL LABELLING OF A CLASS OF TREES." International Journal on Applications of Graph Theory in Wireless Ad hoc Networks and Sensor Networks(GRAPH-HOC) 7, no. 4 (2019): 1–7. https://doi.org/10.5281/zenodo.3361124.

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A labelling or numbering of a graph G with q edges is an assignment of labels to the vertices of G that induces for each edge uv a labelling depending on the vertex labels f(u) and f(v). A labelling is called a graceful labelling if there exists an injective function f: V (G) → {0, 1,2,......q} such that for each edge xy, the labelling │f(x)-f(y)│is distinct. In this paper, we prove that a class of Tn trees are even graceful.

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20

P, Sumathi, and Geetha Ramani G. "Arithmetic Sequential Graceful Labeling of Star Related Graphs – II." Indian Journal of Science and Technology 16, no. 44 (2023): 4038–47. https://doi.org/10.17485/IJST/v16i44.2058.

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Abstract <strong>Objectives:</strong> To identify a new family of arithmetic sequential graceful graphs. <strong>Methods :</strong> The methodology involves mathematical formulation for labeling of the vertices of a given graph and subsequently establishing that these formulations give rise to arithmetic sequential graceful labeling. <strong>Findings:</strong> Here, we introduce the square graph , which shares its vertex set with G. In , two vertices are considered adjacent if their distance in is either 1 or 2. To further elucidate, we identify the splitting graph as

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21

Sethuraman, G., P. Ragukumar, and Peter J. Slater. "Any Tree withmedges can be embedded in a Graceful Tree with less than4medges and in a graceful planar graph." Discrete Mathematics 340, no. 2 (2017): 96–106. http://dx.doi.org/10.1016/j.disc.2016.07.009.

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22

El-Zohny, H., S. Radwan, S. I. Abo El-Fotooh, and Z. Mohammed. "Graceful Labeling of Hypertrees." Journal of Mathematics Research 13, no. 1 (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&nbsp; (vertex labeling) or to edges of G&nbsp; (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&rsquo; 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 (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&nbsp; (vertex labeling) or to edges of G&nbsp; (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&rsquo; labels of each edge is distinct and each one is the label of the corresponding edge.

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Sethuraman, G., and P. Ragukumar. "Towards Optimal Embedding of an Arbitrary Tree in a Graceful Tree." Electronic Notes in Discrete Mathematics 48 (July 2015): 73–80. http://dx.doi.org/10.1016/j.endm.2015.05.011.

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25

Virk, Abaid ur Rehman, and A. Riasat. "Odd Graceful Labeling of W -Tree W T ( n , k ) and its Disjoint Union." Utilitas Mathematica 118 (January 8, 2024): 51–62. http://dx.doi.org/10.61091/um118-05.

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Let G=(V(G),E(G)) be a graph with p vertices and q edges. A graph G of size q is said to be odd graceful if there exists an injection λ:V(G)→0,1,2,…,2q−1 such that assigning each edge xy the label or weight |λ(x)–λ(y)| results in the set of edge labels being 1,3,5,…,2q−1. This concept was introduced in 1991 by Gananajothi. In this paper, we examine the odd graceful labeling of the W-tree, denoted as WT(n,k).

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Sivakumar, S., S. Vidyanandini, D. Haritha, and T. Rajesh Kumar. "CUBIC DIFFERENCE LABELING FOR GRACEFUL TREE CONSTRUCTED FROM CATERPILLAR." Advances in Mathematics: Scientific Journal 9, no. 10 (2020): 8623–27. http://dx.doi.org/10.37418/amsj.9.10.87.

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27

., Yunizar. "PELABELAN GRACEFUL PADA GRAF HALIN G(2, n) UNTUK n ≥ 3." Jurnal Matematika UNAND 3, no. 1 (2014): 89. http://dx.doi.org/10.25077/jmu.3.1.89-92.2014.

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Graf Halin adalah graf planar yang dibangun dari suatu tree T dan suatu cycleC yang menghubungkan setiap titik ujung dari tree. Dalam penelitian ini dikaji tentangpelabelan graceful pada graf Halin G(2, n), untuk n ≥ 3. Pelabelan ini didefinisikanmenjadi dua kasus, yaitu kasus untuk n ganjil dan n ≥ 5, dan kasus untuk n genap dann ≥ 6.

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28

Maharana, Jnanadeva, Sudipta Mukherji, and Sudhakar Panda. "Notes on Axion, Inflation and Graceful Exit in Stringy Cosmology." Modern Physics Letters A 12, no. 07 (1997): 447–56. http://dx.doi.org/10.1142/s0217732397000467.

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We study the classical equations of motion and the corresponding Wheeler–De Witt equations for tree level string effective action with dilaton and axion. The graceful exit problem in certain cases is then analyzed.

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29

Suparta, I. Nengah, and I. Dewa Made Agus Ariawan. "Some methods for constructing some classes of graceful uniform trees." Indonesian Journal of Combinatorics 2, no. 2 (2018): 123. http://dx.doi.org/10.19184/ijc.2018.2.2.7.

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<p>A tree <span class="math"><em>T</em>(<em>V</em>, <em>E</em>)</span> is <span><em>graceful</em></span> if there exists an injective function <span class="math"><em>f</em></span> from the vertex set <span class="math"><em>V</em>(<em>T</em>)</span> into the set <span class="math">{0, 1, 2, ..., ∣<em>V</em>∣ − 1}</span> which induces a bijective function <span class="math"><em>f</em>ʹ</span> from the edge set &l

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30

Krishnaa, A. "A Note On Some Thoughts on the Graceful Tree Conjecture." Journal of Discrete Mathematical Sciences and Cryptography 16, no. 6 (2013): 387–92. http://dx.doi.org/10.1080/09720529.2013.858484.

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31

Auparajita, Krishnaa. "Some Algorithms of Graph Theory in Cryptology." Indian Journal of Advanced Mathematics (IJAM) 4, no. 1 (2024): 9–15. https://doi.org/10.54105/ijam.A1167.04010424.

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<strong>Abstract: </strong>The inventive use of concepts from Graph Theory plays a significant role in hiding the original Plain-text for resulting in a significantly safe data transfer. In this work, the tree traversal algorithms like Inorder, Preorder, Postorder, Kruskal’s algorithm for making minimal spanning tree and the modified graph labelling scheme of graceful labelling allowing repetition of exactly one vertex label for certain graphs, have been employed to create highly hidden Cipher-texts. Encryption and decryption algorithms for all these methods are being presented in this w

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Krishnaa, Auparajita. "Some Algorithms of Graph Theory in Cryptology." Indian Journal of Advanced Mathematics 4, no. 1 (2024): 9–15. http://dx.doi.org/10.54105/ijam.a1167.04010424.

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The inventive use of concepts from Graph Theory plays a significant role in hiding the original Plain-text for resulting in a significantly safe data transfer. In this work, the tree traversal algorithms like Inorder, Preorder, Postorder, Kruskal’s algorithm for making minimal spanning tree and the modified graph labelling scheme of graceful labelling allowing repetition of exactly one vertex label for certain graphs, have been employed to create highly hidden Cipher-texts. Encryption and decryption algorithms for all these methods are being presented in this work.

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33

Simarmata, Nikson, Ikhlas Pratama Sandy, and Kiki A. Sugeng. "Graceful labeling construction for some special tree graph using adjacency matrix." Electronic Journal of Graph Theory and Applications 11, no. 2 (2023): 343. http://dx.doi.org/10.5614/ejgta.2023.11.2.1.

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PANDEY, GEETA BHANDARI. "A Note on Leaf Gall on Alstonia scholaris (Saptaparni) tree due to Infestation of Pauropsylla tuberculata (Gall Insect)." JOURNAL OF ENVIRONMENT AND BIO-SCIENCE 37, no. 02 (2023): 141. http://dx.doi.org/10.59467/jebs.2023.37.141.

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Alstonia scholaris is a perennial evergreen tree of family Apocynaceae and one of the graceful ornamental and medicinal plants. In general, this tree suffers the problem, that is, gall formation on a leaf. Most probably, it is caused by leaf gall insects (Pauropsylla tuberculata) belonging to order Hemiptera and is commonly known as jumping lice. The insect has five nymphal stages and forms the gall by sucking the plant sap. It lacks the pupal stage. Such a heavy attack by this insect affects the rate of photosynthesis and overall growth of plants. This brief note focuses on the development an

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Su, Jing, Qiyue Zhang, and Bing Yao. "The connection between the magical coloring of trees." AIMS Mathematics 9, no. 10 (2024): 27896–907. http://dx.doi.org/10.3934/math.20241354.

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<p>Let $ f $ be a set-ordered edge-magic labeling of a graph $ G $ from $ V(G) $ and $ E(G) $ to $ [0, p-1] $ and $ [1, p-1] $, respectively; it also satisfies the following conditions: $ |f(V(G))| = p $, $ \max f(X) &lt; \min f(Y) $, and $ f(x)+f(y)+f(xy) = C $ for each edge $ xy\in E(G) $. In this paper, we removed the restriction that the labeling of vertices could not be repeated, and presented the concept of magical colorings including edge-magic coloring, edge-difference coloring, felicitous-difference coloring, and graceful-difference coloring. We studied the magical colorings

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George, Reena. "Alone at Christmas." Christian Journal for Global Health 8, no. 1 (2021): 88–89. http://dx.doi.org/10.15566/cjgh.v8i1.499.

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I longedto give you a gift,a pearlof great price.Craftedwithin my wounds and nourishedwith body and blood, the pearl grewlovely and luminous. Tenderly wrapped in gentle huesthe lovely blues of sky and sea,it waited beneath the Christmas tree.Christmas came and went.You were busy. I understand.I always do. The tree and I continued sitting by the window with smiling fairy lights on.Then it was Lent, and I had to put awaythat old tree.The neighbours were sniggering you see.But the pearl and I, we sat waitingfor you, my beloved, Prodigal daughter. And then you came! and I ran,hol

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Xie, Chunping, Guowu Zhang, Chiyung Jim, et al. "Bioclimatic Suitability of Actual and Potential Cultivation Areas for Jacaranda mimosifolia in Chinese Cities." Forests 12, no. 7 (2021): 951. http://dx.doi.org/10.3390/f12070951.

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Jacaranda mimosifolia is regarded as a prized ornamental tree in the urban landscape with attractive, abundant and long-lasting violet-colored flowers and graceful tree form. It has been widely cultivated in recent years in many Chinese cities. However, the lack of scientific and practical guidance to cultivate the exotic species has brought about planting failures in some areas, incurring substantial economic losses and landscape decline. A comprehensive understanding of the current spatial pattern and climatic conditions of J. mimosifolia in China can inform species choice, planting and mana

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38

Davis, Richard H. "Indian Art Objects as Loot." Journal of Asian Studies 52, no. 1 (1993): 22–48. http://dx.doi.org/10.2307/2059143.

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Let us imagine a graceful bronze image of Dancing Śiva before us. It was perhaps created by a Cola artist in eleventh-century Tamilnad to be installed in a temple to receive offerings of worship, and to parade around the town in a ceremonial palanquin on festival days. From there, this image might have followed any of several paths to stand before us now in a North American museum. Perhaps it was buried under a banyan tree in the fourteenth century when invading Islamic armies, feared for their iconoclasm, marched through the Kaveri delta on their way to Madurai. It could have been disinterred

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Narayanan, V., J. B. Veeramalini, and G. Baskar. "Comparative analysis of N-hydroxy-2,6-bis (p-chlorophenyl)-3-isopropylpiperidin-4-one semicarbazone and N-hydroxy-2,6-bis (p-chlorophenyl)-3-isopropylpiperidin-4-one thiosemicarbazone by graceful tree graph, non-graceful graph and cliques representation of graph with the data spectrum." International Journal of Materials and Product Technology 55, no. 1/2/3 (2017): 210. http://dx.doi.org/10.1504/ijmpt.2017.084972.

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Narayanan, V., J. B. Veeramalini, and G. Baskar. "Comparative analysis of N-hydroxy-2,6-bis (p-chlorophenyl)-3-isopropylpiperidin-4-one semicarbazone and N-hydroxy-2,6-bis (p-chlorophenyl)-3-isopropylpiperidin-4-one thiosemicarbazone by graceful tree graph, non-graceful graph and cliques representation of graph with the data spectrum." International Journal of Materials and Product Technology 55, no. 1/2/3 (2017): 210. http://dx.doi.org/10.1504/ijmpt.2017.10005676.

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Suparta, I. Nengah, and I. Dewa M. Agus Ariawan. "Expanding graceful trees." Electronic Journal of Graph Theory and Applications 8, no. 2 (2020): 217. http://dx.doi.org/10.5614/ejgta.2020.8.2.2.

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Hossain, Md Forhad, Md Momin Al Aziz, and M. Kaykobad. "New Classes of Graceful Trees." Journal of Discrete Mathematics 2014 (November 23, 2014): 1–6. http://dx.doi.org/10.1155/2014/194759.

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Graceful labeling is one of the most researched problems. One of the earliest results is that caterpillars are graceful. We show that caterpillars connected to a vertex recursively satisfying certain conditions are also graceful.

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Ragukumar, P., and G. Sethuraman. "Binomial trees are graceful." AKCE International Journal of Graphs and Combinatorics 17, no. 1 (2020): 632–36. http://dx.doi.org/10.1016/j.akcej.2018.06.005.

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Iaboni, Craig, Deepan Lobo, Ji-Won Choi, and Pramod Abichandani. "Event-Based Motion Capture System for Online Multi-Quadrotor Localization and Tracking." Sensors 22, no. 9 (2022): 3240. http://dx.doi.org/10.3390/s22093240.

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Motion capture systems are crucial in developing multi-quadrotor systems due to their ability to provide fast and accurate ground truth measurements for tracking and control. This paper presents the implementation details and experimental validation of a relatively low-cost motion-capture system for multi-quadrotor motion planning using an event camera. The real-time, multi-quadrotor detection and tracking tasks are performed using a deep learning network You-Only-Look-Once (YOLOv5) and a k-dimensional (k-d) tree, respectively. An optimization-based decentralized motion planning algorithm is i

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English, Sean, and Ping Zhang. "On graceful colorings of trees." MATHEMATICA BOHEMICA 142, no. 1 (2016): 57–73. http://dx.doi.org/10.21136/mb.2017.0035-15.

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Suresh Kumar, P., and K. Rajendran. "CORONA PRODUCT OF GRACEFUL TREES." Advances in Mathematics: Scientific Journal 9, no. 10 (2020): 8669–74. http://dx.doi.org/10.37418/amsj.9.10.92.

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Chan, Tsz Lung, Wai Shun Cheung, and Tuen Wai Ng. "Graceful labeling for mushroom trees." Aequationes mathematicae 89, no. 3 (2014): 719–24. http://dx.doi.org/10.1007/s00010-014-0259-5.

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Mishra, Debdas, and Pratima Panigrahi. "Some new classes of graceful Lobsters obtained from diameter four trees." Mathematica Bohemica 135, no. 3 (2010): 257–78. http://dx.doi.org/10.21136/mb.2010.140703.

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Sandy, I. P., A. Rizal, E. N. Manurung, and K. A. Sugeng. "Alternative construction of graceful symmetric trees." Journal of Physics: Conference Series 1008 (April 2018): 012031. http://dx.doi.org/10.1088/1742-6596/1008/1/012031.

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Barrientos, Christian, and Elliot Krop. "Improved Bounds for Relaxed Graceful Trees." Graphs and Combinatorics 33, no. 2 (2017): 287–305. http://dx.doi.org/10.1007/s00373-017-1757-8.

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