Journal articles: 'Double star graph'

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P, Sumathi, and Geetha Ramani G. "Arithmetic Sequential Graceful Labeling on Star Related Graphs." Indian Journal of Science and Technology 15, no. 44 (2022): 2356–62. https://doi.org/10.17485/IJST/v15i44.1863.

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Abstract Objectives: To identify a new family of Arithmetic sequential graceful graphs. Methods: 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. Findings: In this study, we analyzed some star related graphs namely Star graph, Ustar, t-star, and double star proved that these graphs possess Arithmetic sequential graceful labeling. Novelty: Here, we introduced a new labeling called Arithmetic sequential graceful labeling and we give Arithmetic sequential graceful labeling to some star related graphs. Keywords: Star graph; t-star; U-star; Graceful labeling; Double star

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Karthikeyan C. "Various Labelling for Double Star Graph." Journal of Information Systems Engineering and Management 10, no. 18s (2025): 273–77. https://doi.org/10.52783/jisem.v10i18s.2912.

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In this work, we introduce labeling techniques for the double star graph, a basic graph theory structure. In particular, we concentrate on calculating two different kinds of labeling: fortunate labeling and correct labeling. Adhering to the restrictions of vertex labeling in graph theory, appropriate labeling guarantees that neighboring vertices receive unique labels. A more recent and interesting idea is the fortunate labelling, which gives vertices positive numbers so that the labels on neighboring vertices add up to a unique value for each edge. We calculate and examine various labeling methods using algorithmic methods, proving their usefulness and effectiveness for the double star graph. Our findings further enhance the theoretical and practical capabilities of various labeling systems by offering insightful information about how to optimize them for bipartite and tree-like graphs.

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Cao, Jingdong, Shumin Zhang, Chengfu Ye, and Tianxia Jia. "Double-Star Isolation of Maximal Outerplanar Graphs ∗." Journal of Physics: Conference Series 2660, no. 1 (2023): 012023. http://dx.doi.org/10.1088/1742-6596/2660/1/012023.

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Abstract In the graph G = (V, E), V represents the set of vertices and E represents the set of edges. ℱ represents a family of graphs. A subset S ⊆ V is considered an ℱ -isolating set if G[V\NG[S]] does not contain F as a subgraph for all F ∈ ℱ. The ℱ-isolation number of the graph G, denoted by 𝜄(G, ℱ), is defined as the minimum cardinality of an ℱ-isolating set in G. If the family ℱ consists of a single graph H, then the subset S is referred to as an H-isolating set. The H-isolation number of the graph G, denoted by 𝜄 H (G), is defined as the minimum cardinality of an H-isolating set in G. Maximal outerplanar graphs have been widely applied in different fields of research since 1891, where they hold significant importance. This paper aims to provide a comprehensive analysis of the H-isolation number in maximal outerplanar graphs, ι H ( G ) ≤ min { n 2 k + 4 , n + n 2 2 k + 5 , n − n 2 2 k + 2 } , where H ≅ S 1,k+1,k+1.

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Vivin.J, Vernold, Venkatachalam M., and Kaliraj K. "Harmonious coloring on double star graph families." Tamkang Journal of Mathematics 43, no. 2 (2012): 153–58. http://dx.doi.org/10.5556/j.tkjm.43.2012.675.

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In this present paper, we have proved for the line graph of double star graph, the harmonious chromatic number and the achromatic number are equal. As a motivation this work can be extended by classifying the different families of graphs for which these two numbers are equal.

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Venkatacha, M., N. Mohanapriy, and J. Vernold Vivin. "Star Coloring on Double Star Graph Families." Journal of Modern Mathematics and Statistics 5, no. 1 (2011): 33–36. http://dx.doi.org/10.3923/jmmstat.2011.33.36.

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Sugeng, K. A., Z. Z. Barack, N. Hinding, and R. Simanjuntak. "Modular Irregular Labeling on Double-Star and Friendship Graphs." Journal of Mathematics 2021 (December 28, 2021): 1–6. http://dx.doi.org/10.1155/2021/4746609.

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A modular irregular graph is a graph that admits a modular irregular labeling. A modular irregular labeling of a graph G of order n is a mapping of the set of edges of the graph to 1,2 , … , k such that the weights of all vertices are different. The vertex weight is the sum of its incident edge labels, and all vertex weights are calculated with the sum modulo n . The modular irregularity strength is the minimum largest edge label such that a modular irregular labeling can be done. In this paper, we construct a modular irregular labeling of two classes of graphs that are biregular; in this case, the regular double-star graph and friendship graph classes are chosen. Since the modular irregularity strength of the friendship graph also holds the minimal irregularity strength, then the labeling is also an irregular labeling with the same strength as the modular case.

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Shirkol, Shailaja S., Pavitra P. Kumbargoudra, and Meenal M. Kaliwal. "DOUBLE ROMAN DOMINATION NUMBER OF MIDDLE GRAPH." South East Asian J. of Mathematics and Mathematical Sciences 18, no. 03 (2022): 369–80. http://dx.doi.org/10.56827/seajmms.2022.1803.31.

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For any graph G(V, E), a function f : V (G) 0, 1, 2, 3 is called Double Roman dominating function (DRDF) if the following properties holds, If f (v) = 0, then there exist two vertices v1, v2 ∈ N (v) for which f (v1) = f (v2) = 2 or there exist one vertex u ∈ N (v) for which f (u) = 3.∈ If f (v) = 1, then there exist one vertex u N (v) for which f (u) = 2 or Σ f (u) = 3. The weight of DRDF is the value w(f ) = v∈V (G) f (v). The minimum weight among all double Roman dominating function is called double Roman domination number and is denoted by γdR(G). In this article we initiated research on double Roman domination number for middle graphs. We established lower and upper bounds and also we characterize the double Roman domination number of middle graphs. Later we calculated numerical value of double Roman domination number of middle graph of path, cycle, star, double star and friendship graphs.

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P., Kavitha*1 &. A. Rajasekaran2. "PRIME LABELING IN DUPLICATE GRAPH OF SOME GRAPHS." GLOBAL JOURNAL OF ENGINEERING SCIENCE AND RESEARCHES 6, no. 2 (2019): 35–45. https://doi.org/10.5281/zenodo.2558187.

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A graph with vertices is said to admit prime labeling if its vertices can be labeled with distinct positive integers not exceeding such that the labels of each pair of adjacent vertices are relatively prime. A graph which admits prime labeling is called a prime graph. In this paper we prove that the duplicate graph of the path the duplicate graph of cycle the duplicate graph of star  the duplicate graph of double star  the duplicate graph of comb graph  and the duplicate graph of bistar graph for all integers  are prime labeling

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Senthamizh selvan J and Jahir Hussain R. "Coefficient of Range Labeling of Some Graphs." Journal of Computational Mathematica 7, no. 2 (2023): 044–56. http://dx.doi.org/10.26524/cm175.

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The coefficient of range labeling is a new type of labeling since it was introduced in 2022 by Senthamizh Selvan and Jahir Hussain. In this paper, we obtained a coefficient of range labeling for the path graph, cycle graph, sun graph, and double star graph is coefficient of range graph. We can also determine the coefficient of range value of the path graph, cycle graph, sun graph, and double star graph.

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Kavitha, K., and N. G. David. "Dominator Coloring on Star and Double Star Graph Families." International Journal of Computer Applications 48, no. 3 (2012): 22–25. http://dx.doi.org/10.5120/7328-0185.

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Aytaç, Aysun, and Ayşen Mutlu. "Double domination number of the shadow (2,3)-distance graphs." Malaya Journal of Matematik 11, no. 02 (2023): 228–38. http://dx.doi.org/10.26637/mjm1102/011.

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Let G (V,E) be a graph with the vertex set V (G) and S be a subset of V(G). If every vertex of V is dominated by S at least twice, then the set S is called a double domination set of the graph. The number of elements of the double domination set with the smallest cardinality is called double domination number and denoted by 2 (G)  notation. In this paper, we discussed the double domination parameter on some types of shadow distance graphs such as cycle, path, star, complete bipartite and wheel graphs.

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12

Kusumastuti, Nilamsari, Raventino, and Fransiskus Fran. "The diachromatic number of double star graph." Journal of Physics: Conference Series 2106, no. 1 (2021): 012024. http://dx.doi.org/10.1088/1742-6596/2106/1/012024.

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Abstract We are interested in the extension for the concept of complete colouring for oriented graph G → that has been proposed in many different notions by several authors (Edwards, Sopena, and Araujo-Pardo in 2013, 2014, and 2018, respectively). An oriented colouring is complete if for every ordered pair of colours, at least one arc in G → whose endpoints are coloured with these colours. The diachromatic number, dac ( G → ) , is the greatest number of colours in a complete oriented colouring. In this paper, we establish the formula of diachromatic numbers for double star graph, k 1 , n , n → , over all possible orientations on the graph. In particular, if din (u) = 0 (resp. dout(u) = 0)and din (wi ) = 1 (resp. dout (w 1) = 1) for all i, then dac ( k 1 , n , n → ) = ⌊ n ⌋ + 1 , where u is the internal vertex and w i , i ∈ {1,…, n}, is the pendant vertices of the digraph.

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Arundhadhi, R., and R. Sattanathan. "Acyclic Coloring on Double Star Graph Families." International Journal of Computer Applications 42, no. 18 (2012): 32–35. http://dx.doi.org/10.5120/5794-8131.

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R., Jahir Hussain, and Senthamizh Selvan J. "RANGE LABELING FOR SOME GRAPHS." INTERNATIONAL JOURNAL OF MATHEMATICS AND COMPUTER RESEARCH 11, no. 05 (2023): 3392–95. https://doi.org/10.5281/zenodo.7895916.

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15

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 berlian rangkap berbintang, kelas graf pohon, graf K_(n,n) ⨀ K_1. Labeling of a graph is a mapping that brings every graph element, namely the edge or vertex, to the positive integers, which is called label. A function f is called graceful labeling of graph G with m edge if is injective and induced function defined as is bijective. A graph that has graceful labeling is called a graceful graph. The construction of graceful labeling in the double-star diamond graph , some classes of tree graphs, and certain corona graph (K_(n,n) ⨀ K_1) will be shown in this paper. Keywords: Graceful labeling, double-star diamond graph, class of tree graph, K_(n,n) ⨀ K_1 graph.  

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P, Kowsalya, and Vijayalakshmi D. "On Radio k – Chromatic Number for Mycielski of Some Graphs." Indian Journal of Science and Technology 17, no. 39 (2024): 4129–37. https://doi.org/10.17485/IJST/v17i39.2315.

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Abstract Objectives: In radio networks, the main difficulty is managing the radio spectrum, assigning radio frequencies to transmitters optimally without any interferences. This study aims to find the smallest span of Mycielski of some graphs, the maximum color assigned to any node is called span. Methods : This study focused on the problem of reducing interference by modeling it with a radio k - coloring problem on graphs. Where transmitters are modeled as nodes in a graph, with edges connecting nodes that represent transmitters in close proximity to one another. For a graph , with node set , edge set and an integer , a radio k - coloring of is a function satisfying the condition for any two nodes and , where is the distance between and in . The radio k - coloring of , is the maximum color assigned to any node of and it is denoted by . The radio k -chromatic number of is the minimum value of taken over all radio k - coloring of and it is denoted by . Findings: This study obtained the radio k-chromatic number for Mycielski of some graphs for and 3 Novelty: To solve the channel assignment problem in radio transmitters, the interference graph is developed, and the channel assignment has been converted into a graph coloring. Reducing the interference by a radio k - coloring problem will motivate many researchers to find the radio k - coloring in various graphs. Keywords: Radio k – Coloring, Mycielski graph, Double Star graph, Triple Star graph, Sunlet graph, Helm graph and Closed Helm graph

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Alraqad, Tariq A., Igor Ž. Milovanović, Hicham Saber, Akbar Ali, Jaya P. Mazorodze, and Adel A. Attiya. "Minimum atom-bond sum-connectivity index of trees with a fixed order and/or number of pendent vertices." AIMS Mathematics 9, no. 2 (2024): 3707–21. http://dx.doi.org/10.3934/math.2024182.

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<abstract>Let $ d_u $ be the degree of a vertex $ u $ of a graph $ G $. The atom-bond sum-connectivity (ABS) index of a graph $ G $ is the sum of the numbers $ (1-2(d_v+d_w)^{-1})^{1/2} $ over all edges $ vw $ of $ G $. This paper gives the characterization of the graph possessing the minimum ABS index in the class of all trees of a fixed number of pendent vertices; the star is the unique extremal graph in the mentioned class of graphs. The problem of determining graphs possessing the minimum ABS index in the class of all trees with $ n $ vertices and $ p $ pendent vertices is also addressed; such extremal trees have the maximum degree $ 3 $ when $ n\ge 3p-2\ge7 $, and the balanced double star is the unique such extremal tree for the case $ p = n-2 $.</abstract>

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Yu, Ber-Lin, Zhongshan Li, Gu-Fang Mou, and Sanzhang Xu. "Eventual Positivity of a Class of Double Star-like Sign Patterns." Symmetry 14, no. 3 (2022): 512. http://dx.doi.org/10.3390/sym14030512.

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Identifying and classifying the potentially eventually positive sign patterns and the potentially eventually exponentially positive sign patterns of orders greater than 3 have been raised as two open problems since 2010. In this article, we investigate the potential eventual positivity of the class of double star-like sign patterns S(n,m,1) whose underlying graph G(S(n,m,1)) is obtained from the underlying graph G(S(n,m)) of the (n+m)-by-(n+m) double star sign patterns S(n,m) by adding an additional vertex adjacent to the two center vertices and removing the edge between the center vertices. We firstly establish some necessary conditions for a double star-like sign pattern to be potentially eventually positive, and then identify all the minimal potentially eventually positive double star-like sign patterns. Secondly, we classify all the potentially eventually positive sign patterns in the class of double star-like sign patterns S(n,m,1). Finally, as an application of our results about the potentially eventually positive double star-like sign patterns, we identify all the minimal potentially eventually exponentially positive sign patterns and characterize all the potentially eventually exponentially positive sign patterns in the class of double star-like sign patterns S(n,m,1).

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G.Nirmala, K.Anitha and S.Rajarajeswari. "Algorithm for Finding Roman Dominating Number of Extended Duplicate Graph of Star Families." February 2023 9, no. 02 (2023): 11–13. http://dx.doi.org/10.46501/ijmtst0902002.

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20

Pati, Sadashiba, Vinay Singh, and Nibedita Adhikari. "On Graph Theoretical Properties of Extended Double Star Interconnection Network Topology." Journal of Advanced Zoology 44, S-3 (2023): 1323–35. http://dx.doi.org/10.17762/jaz.v44is-3.1642.

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 The Extended Double star (EDS) parallel interconnection network with a network controller (NC) is a two-level hybrid structure. It is a large-scale network with the Double star as its basic building block. EDS network has degree (n!+n+1) and diameter ⌊(

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21

Bahadır, Selim. "Graphs with Total Domination Number Double of the Matching Number." Journal of New Theory, no. 49 (December 31, 2024): 1–6. https://doi.org/10.53570/jnt.1520557.

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A subset $S$ of vertices of a graph $G$ with no isolated vertex is called a total dominating set of $G$ if each vertex of $G$ has at least one neighbor in the set $S$. The total domination number $\gamma_t(G)$ of a graph $G$ is the minimum value of the size of a total dominating set of $G$. A subset $M$ of the edges of a graph $G$ is called a matching if no two edges of $M$ have a common vertex. The matching number $\nu (G)$ of a graph $G$ is the maximum value of the size of a matching in $G$. It can be observed that $\gamma_t(G)\leq 2\nu(G)$ holds for every graph $G$ with no isolated vertex. This paper studies the graphs satisfying the equality and proves that $\gamma_t(G)= 2\nu(G)$ if and only if every connected component of $G$ is either a triangle or a star.

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Kowsalya, P., and D. Vijayalakshmi. "On Radio k – Chromatic Number for Mycielski of Some Graphs." Indian Journal Of Science And Technology 17, no. 39 (2024): 4129–37. http://dx.doi.org/10.17485/ijst/v17i39.2315.

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Objectives: In radio networks, the main difficulty is managing the radio spectrum, assigning radio frequencies to transmitters optimally without any interferences. This study aims to find the smallest span of Mycielski of some graphs, the maximum color assigned to any node is called span. Methods : This study focused on the problem of reducing interference by modeling it with a radio k - coloring problem on graphs. Where transmitters are modeled as nodes in a graph, with edges connecting nodes that represent transmitters in close proximity to one another. For a graph , with node set , edge set and an integer , a radio k - coloring of is a function satisfying the condition for any two nodes and , where is the distance between and in . The radio k - coloring of , is the maximum color assigned to any node of and it is denoted by . The radio k -chromatic number of is the minimum value of taken over all radio k - coloring of and it is denoted by . Findings: This study obtained the radio k-chromatic number for Mycielski of some graphs for and 3 Novelty: To solve the channel assignment problem in radio transmitters, the interference graph is developed, and the channel assignment has been converted into a graph coloring. Reducing the interference by a radio k - coloring problem will motivate many researchers to find the radio k - coloring in various graphs. Keywords: Radio k – Coloring, Mycielski graph, Double Star graph, Triple Star graph, Sunlet graph, Helm graph and Closed Helm graph

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T. Lavanya. "Extended Reverse R Degrees of Vertices and Extended Reverse R indices of Graphs." Communications on Applied Nonlinear Analysis 31, no. 5s (2024): 110–17. http://dx.doi.org/10.52783/cana.v31.1005.

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A topological representation of a molecule is called molecular graph. A molecular graph is a collection of points representing the atoms in the molecule and set of lines represent the covalent bonds. Topological indices gather data from the graph of molecule and help to foresee properties of the concealing molecule. All the degree based topological indices have been defined through classical degree concept. In this paper, we define a novel degree concept for a vertex of a simple connected graph: Extended Reverse R degree and also, we define Extended Reverse R indices of a simple connected graph by using the Extended Reverse R degree concept. We compute the Extended Reverse R indices using the above contemporary degree concept for well-known simple connected graphs such as complete bipartite graph, Wheel graph, Generalized Peterson graph, Crown graph, Double star graph, and Windmill graph.

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K.Sunitha. "Radial Radio Pell Mean Labeling of Subdivision of Graphs." Advances in Nonlinear Variational Inequalities 28, no. 2 (2024): 267–73. http://dx.doi.org/10.52783/anvi.v28.1969.

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A one-one mapping ϕ : V (G) → N for a connected graph G is defined as follows: d(x,y)+⌈(ϕ(x)+2ϕ(y))/2⌉≥1+r(G), where radius is denoted by r(G). Any vertex in G has a radial radio pell mean number of ϕ which is the maximum number and is represented by rrpmn(ϕ). Here, we look at the labeling of various graphs using the radial radio pell mean of subdivisions such as subdivision of star graph S(K_(1,n)) , subdivision of path graph S(Pn) , subdivision of friendship graph S(Fn), subdivision of wheel graph S(Wn), subdivision of quadrilateral book graph S(QB(4,3)), subdivision of closed helm graph S(CHn), subdivision of helm graph S(Hn), subdivision of double fan graph S(DFn) and subdivision of triangular book graph S(TB(3,n)).

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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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Kozerenko, Sergiy, and Andrii Serdiuk. "New results on imbalance graphic graphs." Opuscula Mathematica 43, no. 1 (2023): 81–100. http://dx.doi.org/10.7494/opmath.2023.43.1.81.

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An edge imbalance provides a local measure of how irregular a given graph is. In this paper, we study graphs with graphic imbalance sequences. We give a new proof of imbalance graphicness for trees and use the new idea to prove that the same holds for unicyclic graphs. We then show that antiregular graphs are imbalance graphic and consider the join operation on graphs as well as the double graph operation. Our main results are concerning imbalance graphicness of three classes of block graphs: block graphs having all cut vertices in a single block; block graphs in which the subgraph induced by the cut vertices is either a star or a path. In the end, we discuss open questions and conjectures regarding imbalance graphic graphs.

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Yang, Chen, and Chongmin Li. "Graph Entropy Based on Wiener Polarity Index Under Four Kinds of Graph Operations." Mathematics and Computer Science 10, no. 1 (2025): 19–25. https://doi.org/10.11648/j.mcs.20251001.13.

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The Wiener polarity index of a graph &lt;i&gt;G&lt;/i&gt;, is the number of unordered pairs of vertices that are at distance 3 in &lt;i&gt;G&lt;/i&gt;. This index can reflect the specific distance relation between vertices in the graph, and provides a new way for the study of graph structure. In this paper, the graph entropy based on Wiener polarity index defined. Based on the above definition of graph entropy, it compares the graph entropy of path and balanced double star graphs based on Wiener polarity index. The expressions of graph entropy based on Wiener polarity index for trees with diameter &lt;i&gt;d ≥ 3&lt;/i&gt; are studied under four graph operations: tensor product, strong product, Cartesian product and composite graph.

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Wang, Cheng. "The effect of planetary gear/star gear on the transmission efficiency of closed differential double helical gear train." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 234, no. 21 (2020): 4215–23. http://dx.doi.org/10.1177/0954406220921205.

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The research of transmission efficiency is of great significance for reducing energy consumption and improving the performance of the device. Researchers have done a lot of work on the calculation of transmission efficiency. However, in the present research work, the quantity of planetary gear/star gear is usually not considered and only a planetary gear/star gear is adopted in the gear transmission efficiency. In practice, in order to increase the stiffness and load capacity of gear train, a plurality of planetary gear/star gears is adopted. The closed differential double helical gear train has been widely used in many fields, such as the main reducer of aircraft engine, lifting mechanism, and the power transmission system of marine ships. Therefore, in this paper, the closed differential double helical gear train is taken as the research object and the effect of planetary gear/star gear on the transmission efficiency is analyzed. Firstly, according to the structure of closed differential double helical gear train, related kinematic analysis is given. Secondly, a graph representation is used to characterize the closed differential double helical gear train. According to the theory of virtual power, the power flow direction of closed differential double helical gear train is determined and the value of split power is obtained. According to the input and output values described in graph representation of closed differential double helical gear train, the formula of transmission efficiency is derived and the effects of planetary gear/star gear on transmission efficiency are analyzed. Finally, an illustrative example shows that compared with the theoretical value, the difference considering the effect of planetary gear/star gear on the transmission efficiency of closed differential double helical gear train is two percentage points.

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Das, Kinkar Chandra. "On the Exponential Atom-Bond Connectivity Index of Graphs." Mathematics 13, no. 2 (2025): 269. https://doi.org/10.3390/math13020269.

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Several topological indices are possibly the most widely applied graph-based molecular structure descriptors in chemistry and pharmacology. The capacity of topological indices to discriminate is a crucial component of their study. In light of this, the literature has introduced the exponential vertex-degree-based topological index. The exponential atom-bond connectivity index is defined as follows: eABC=eABC(Υ)=∑vivj∈E(Υ)edi+dj−2didj, where di is the degree of the vertex vi in Υ. In this paper, we prove that the double star DSn−3,1 is the second maximal graph with respect to the eABC index of trees of order n. We give an upper bound on eABC of unicyclic graphs of order n and characterize the maximal graphs. The graph K1∨(P3∪(n−4)K1) gives the maximal graph with respect to the eABC index of bicyclic graphs of order n. We present several relations between eABC(Υ) and ABC(Υ) of graph Υ. Finally, we provide a conclusion summarizing our findings and discuss potential directions for future research.

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Lisna, P. C., and M. S. Sunitha. "b-Chromatic sum of a graph." Discrete Mathematics, Algorithms and Applications 07, no. 04 (2015): 1550040. http://dx.doi.org/10.1142/s1793830915500408.

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A b-coloring of a graph G is a proper coloring of the vertices of G such that there exists a vertex in each color class joined to at least one vertex in each other color classes. The b-chromatic number of a graph G, denoted by [Formula: see text], is the maximum integer [Formula: see text] such that G admits a b-coloring with [Formula: see text] colors. In this paper we introduce a new concept, the b-chromatic sum of a graph [Formula: see text], denoted by [Formula: see text] and is defined as the minimum of sum of colors [Formula: see text] of [Formula: see text] for all [Formula: see text] in a b-coloring of [Formula: see text] using [Formula: see text] colors. Also obtained the b-chromatic sum of paths, cycles, wheel graph, complete graph, star graph, double star graph, complete bipartite graph, corona of paths and corona of cycles.

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Rücker, Gerta, Christoph Rücker, and Ivan Gutman. "On Kites, Comets, and Stars. Sums of Eigenvector Coefficients in (Molecular) Graphs." Zeitschrift für Naturforschung A 57, no. 3-4 (2002): 143–53. http://dx.doi.org/10.1515/zna-2002-3-406.

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Two graph invariants were encountered that form the link between (molecular) walk counts and eigenvalues of graph adjacency matrices. In particular, the absolute value of the sum of coefficients of the first or principal (normalized) eigenvector, s1, and the analogous quantity sn, pertaining to the last eigenvector, appear in equations describing some limits (for infinitely long walks) of relative frequencies of several walk counts. Quantity s1 is interpreted as a measure of mixedness of a graph, and sn, which plays a role for bipartite graphs only, is interpreted as a measure of the imbalance of a bipartite graph. Consequently, sn is maximal for star graphs, while the minimal value of sn is zero. Mixedness s1 is maximal for regular graphs. Minimal values of s1 were found by exhaustive computer search within the sample of all simple connected undirected n-vertex graphs, n≤10: They are encountered among graphs called kites. Within the special sample of tree graphs (searched for n≤20) so-called double snakes have maximal s1, while the trees with minimal s1 are so-called comets. The behaviour of stars and double snakes can be described by exact equations, while approximate equations for s1 of kites and comets could be derived that are fully compatible with and allow to predict some pecularities of the results of the computer search. Finally, the discriminating power of s1, determined within trees and 4-trees (alkanes), was found to be high.

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Xu, Xin, Xu Zhang, and Jiawei Shao. "Planar Turán number of double star $ S_{3, 4} $." AIMS Mathematics 10, no. 1 (2025): 1628–44. https://doi.org/10.3934/math.2025075.

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Planar Turán number, denoted by $ \mathrm{ex}_{\mathcal{P}}(n, H) $, is the maximum number of edges in an $ n $-vertex planar graph which does not contain $ H $ as a subgraph. Ghosh, Győri, Paulos and Xiao initiated the topic of the planar Turán number for double stars. There were two double stars $ S_{3, 4} $ and $ S_{3, 5} $ that remained unknown. In this paper, we give the exact value of $ \mathrm{ex}_{\mathcal{P}}(n, S_{3, 4}) $.

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Priyadharshini, V., and M. Nalliah. "Local distance antimagic chromatic number for the union of star and double star graphs." Ukrains’kyi Matematychnyi Zhurnal 75, no. 5 (2023): 669–82. http://dx.doi.org/10.37863/umzh.v75i5.7075.

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UDC 519.17 Let G = ( V , E ) be a graph on p vertices with no isolated vertices. A bijection f from V to { 1,2 ,3 , … , p } is called a local distance antimagic labeling if, for any two adjacent vertices u and v , we receive distinct weights (colors), where a vertex x has the weight w ( x ) = ∑ v ϵ N ( x ) f ( v ) . The local distance antimagic chromatic number χ l ⅆ a ( G ) is defined as the least number of colors used in any local distance antimagic labeling of G . We determine the local distance antimagic chromatic number for the disjoint union of t copies of stars and double stars.

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Lucci, Massimiliano, Davide Cassi, Vittorio Merlo, Roberto Russo, Gaetano Salina, and Matteo Cirillo. "Josephson Currents and Gap Enhancement in Graph Arrays of Superconductive Islands." Entropy 23, no. 7 (2021): 811. http://dx.doi.org/10.3390/e23070811.

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Evidence is reported that topological effects in graph-shaped arrays of superconducting islands can condition superconducting energy gap and transition temperature. The carriers giving rise to the new phase are couples of electrons (Cooper pairs) which, in the superconducting state, behave as predicted for bosons in our structures. The presented results have been obtained both on star and double comb-shaped arrays and the coupling between the islands is provided by Josephson junctions whose potential can be tuned by external magnetic field or temperature. Our peculiar technique for probing distribution on the islands is such that the hopping of bosons between the different islands occurs because their thermal energy is of the same order of the Josephson coupling energy between the islands. Both for star and double comb graph topologies the results are in qualitative and quantitative agreement with theoretical predictions.

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Shahzad, Muhammad, Muhammad Ahsan Asim, Roslan Hasni, and Ali Ahmad. "Computing Edge Irregularity Strength of Star and Banana Trees Using Algorithmic Approach." Ars Combinatoria 159, no. 1 (2024): 11–20. http://dx.doi.org/10.61091/ars159-02.

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After the Chartrand definition of graph labeling, since 1988 many graph families have been labeled through mathematical techniques. A basic approach in those labelings was to find a pattern among the labels and then prove them using sequences and series formulae. In 2018, Asim applied computer-based algorithms to overcome this limitation and label such families where mathematical solutions were either not available or the solution was not optimum. Asim et al. in 2018 introduced the algorithmic solution in the area of edge irregular labeling for computing a better upper-bound of the complete graph $es(K_n)$ and a tight upper-bound for the complete $m$-ary tree ${es(T}_{m,h})$ using computer-based experiments. Later on, more problems like complete bipartite and circulant graphs were solved using the same technique. Algorithmic solutions opened a new horizon for researchers to customize these algorithms for other types of labeling and for more complex graphs. In this article, to compute edge irregular $k$-labeling of star $S_{m,n}$ and banana tree ${BT}_{m,n}$, new algorithms are designed, and results are obtained by executing them on computers. To validate the results of computer-based experiments with mathematical theorems, inductive reasoning is adopted. Tabulated results are analyzed using the law of double inequality and it is concluded that both families of trees observe the property of edge irregularity strength and are tight for $\left\lceil \frac{|V|}{2} \right\rceil$-labeling.

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Septory, Brian Juned, Liliek Susilowati, Dafik Dafik, and Veerabhadraiah Lokesha. "On the Study of Rainbow Antimagic Connection Number of Comb Product of Friendship Graph and Tree." Symmetry 15, no. 1 (2022): 12. http://dx.doi.org/10.3390/sym15010012.

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Given a graph G with vertex set V(G) and edge set E(G), for the bijective function f(V(G))→{1,2,⋯,|V(G)|}, the associated weight of an edge xy∈E(G) under f is w(xy)=f(x)+f(y). If all edges have pairwise distinct weights, the function f is called an edge-antimagic vertex labeling. A path P in the vertex-labeled graph G is said to be a rainbow x−y path if for every two edges xy,x′y′∈E(P) it satisfies w(xy)≠w(x′y′). The function f is called a rainbow antimagic labeling of G if there exists a rainbow x−y path for every two vertices x,y∈V(G). We say that graph G admits a rainbow antimagic coloring when we assign each edge xy with the color of the edge weight w(xy). The smallest number of colors induced from all edge weights of antimagic labeling is the rainbow antimagic connection number of G, denoted by rac(G). This paper is intended to investigate non-symmetrical phenomena in the comb product of graphs by considering antimagic labeling and optimizing rainbow connection, called rainbow antimagic coloring. In this paper, we show the exact value of the rainbow antimagic connection number of the comb product of graph Fn⊳Tm, where Fn is a friendship graph with order 2n+1 and Tm∈{Pm,Sm,Brm,p,Sm,m}, where Pm is the path graph of order m, Sm is the star graph of order m+1, Brm,p is the broom graph of order m+p and Sm,m is the double star graph of order 2m+2.

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Monikandan, S., and S. Sundar Raj. "Adversary degree associated reconstruction number of graphs." Discrete Mathematics, Algorithms and Applications 07, no. 01 (2015): 1450069. http://dx.doi.org/10.1142/s1793830914500694.

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A vertex-deleted subgraph of a graph G is called a card of G. A card of G with which the degree of the deleted vertex is also given is called a degree associated card or dacard of G. The adversary degree associated reconstruction number of a graph G, adrn (G), is the minimum number k such that every collection of k dacards of G uniquely determines G. We prove that adrn (G) = 1 + min {t+1, m-t} or 1 + min {t, m - t + 2} for a graph G obtained by subdividing t edges of K1, m. We also prove that if G is a nonempty disconnected graph whose components are cycles or complete graphs, then adrn (G) is 3 or 4, while, if G is a double star whose central vertices have degrees m + 1 and n + 1(m > n ≥ 2), then adrn (G) can be as large as n + 3.

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Ulaganathan, P. P., B. Selvam, and P. Vijaya kumar. "Signed Product Cordial labeling in duplicate graphs of Bistar, Double Star and Triangular Ladder Graph." International Journal of Mathematics Trends and Technology 33, no. 1 (2016): 19–24. http://dx.doi.org/10.14445/22315373/ijmtt-v33p505.

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Marsidi, Ika Hesti Agustin, Dafik, Elsa Yuli Kurniawati, and Rosanita Nisviasari. "The rainbow vertex antimagic coloring of tree graphs." Journal of Physics: Conference Series 2157, no. 1 (2022): 012019. http://dx.doi.org/10.1088/1742-6596/2157/1/012019.

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Abstract Let G(V (G),E(G)) be a connected, simple, and finite graph. Let f be a bijective function of labeling on graph G from the edge set E(G) to natural number up to the number of edges of G. A rainbow vertex antimagic labeling of graph G is a function f under the condition all internal vertices of a path u – υ, Ɐu, υ ∈ V (G) have different weight (denoted by w(u)), where w(u) = ∑ uu′∈E(G)f (uu′). If G has a rainbow vertex antimagic labeling, then G is a rainbow vertex antimagic coloring, where the every vertex is assigned with the color w(u). The rvac(G) is a notation of rainbow vertex antimagic connection number of graph G which means the minimum colors taken over all rainbow vertex antimagic coloring induced by rainbow vertex antimagic labeling of graph G. The results of this research are the exact value of the rainbow vertex antimagic connection number of star (Sn ), double star (DSn ), and broom graph (Brn, m ).

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40

Kristiana, Arika Indah, M. Hidayat, Robiatul Adawiyah, D. Dafik, Susi Setiawani, and Ridho Alfarisi. "ON LOCAL IRREGULARITY OF THE VERTEX COLORING OF THE CORONA PRODUCT OF A TREE GRAPH." Ural Mathematical Journal 8, no. 2 (2022): 94. http://dx.doi.org/10.15826/umj.2022.2.008.

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Let $G=(V,E)$ be a graph with a vertex set $V$ and an edge set $E$. The graph $G$ is said to be with a local irregular vertex coloring if there is a function $f$ called a local irregularity vertex coloring with the properties: (i) $l:(V(G)) \to \{ 1,2,...,k \} $ as a vertex irregular $k$-labeling and $w:V(G)\to N,$ for every $uv \in E(G),$ ${w(u)\neq w(v)}$ where $w(u)=\sum_{v\in N(u)}l(i)$ and (ii) $\mathrm{opt}(l)=\min\{ \max \{ l_{i}: l_{i} \ \text{is a vertex irregular labeling}\}\}$. The chromatic number of the local irregularity vertex coloring of $G$ denoted by $\chi_{lis}(G)$, is the minimum cardinality of the largest label over all such local irregularity vertex colorings. In this paper, we study a local irregular vertex coloring of $P_m\bigodot G$ when $G$ is a family of tree graphs, centipede $C_n$, double star graph $(S_{2,n})$, Weed graph $(S_{3,n})$, and $E$ graph $(E_{3,n})$.

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Lourdusamy, A., S. Kither Iammal, and I. Dhivviyanandam. "Monophonic Cover Pebbling Number $(MCPN)$ of Network Graphs." Utilitas Mathematica 121, no. 1 (2024): 11–24. https://doi.org/10.61091/um121-02.

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Given a connected graph $G$ and a configuration $D$ of pebbles on the vertices of $G$, a pebbling transformation involves removing two pebbles from one vertex and placing one pebble on its adjacent vertex. A monophonic path is defined as a chordless path between two non-adjacent vertices $u$ and $v$. The monophonic cover pebbling number, $\gamma_{\mu}(G)$, is the minimum number of pebbles required to ensure that, after a series of pebbling transformations using monophonic paths, all vertices of $G$ are covered with at least one pebble each. In this paper, we determine the monophonic cover pebbling number ($MCPN$) for the gear graph, sunflower planar graph, sun graph, closed sun graph, tadpole graph, lollipop graph, double star-path graph, and a class of fuses.

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42

Kristiana, Arika Indah, Sri Moeliyana Citra, Dafik, Ridho Alfarisi, and Robiatul Adawiyah. "On The Packing k-Coloring of Some Family Trees." Statistics, Optimization & Information Computing 13, no. 3 (2024): 1291–98. https://doi.org/10.19139/soic-2310-5070-2047.

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All graphs in this paper are simple and connected. Let $G=(V,E)$ be a graph where $V(G)$ is nonempty of vertex set of $G$ and $E(G)$ is possibly empty set of unordered pairs of elements of $V(G)$. The distance from $u$ to $v$ in $G$ is the length of a shortest path joining them, denoted by $d(u,v)$. For some positive integer $k$, a function $ c:V(G)\rightarrow \{1,2,...k\} $ is called packing $k-$coloring if any two not adjacent vertices $u$ and $v$, $c(u)=c(v)=i$ and $d(u,v)\geq i+1$. The minimum number $k$ such that the graph $G$ has a packing $k-$coloring is called the packing chromatic number, denoted by $\chi_\rho(G) $. In this paper, we investigate the packing chromatic number of some family trees, namely centipede, firecracker, broom, double star and banana tree graphs.

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43

Chartrand, Gary, and Ping Zhang. "The Ascending Ramsey Index of a Graph." Symmetry 15, no. 2 (2023): 523. http://dx.doi.org/10.3390/sym15020523.

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Let G be a graph with a given red-blue coloring c of the edges of G. An ascending Ramsey sequence in G with respect to c is a sequence G1, G2, …, Gk of pairwise edge-disjoint subgraphs of G such that each subgraph Gi (1≤i≤k) is monochromatic and Gi is isomorphic to a proper subgraph of Gi+1 (1≤i≤k−1). The ascending Ramsey index ARc(G) of G with respect to c is the maximum length of an ascending Ramsey sequence in G with respect to c. The ascending Ramsey index AR(G) of G is the minimum value of ARc(G) among all red-blue colorings c of G. It is shown that there is a connection between this concept and set partitions. The ascending Ramsey index is investigated for some classes of highly symmetric graphs such as complete graphs, matchings, stars, graphs consisting of a matching and a star, and certain double stars.

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Rajesh Kumar., T. J. "Total neighborhood prime labeling of some trees." Proyecciones (Antofagasta) 41, no. 1 (2022): 101–10. http://dx.doi.org/10.22199/issn.0717-6279-4819.

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Let G be a graph with p vertices and q edges. A total neighborhood prime labeling of G is a labeling in which the vertices and edges are assigned labels from 1 to p + q such that the gcd of labeling in the neighborhood of each non degree 1 vertex is equal to 1 and the gcd of labeling in the edges of each non degree 1 vertex is equal to 1. A graph that admits a total neighborhood prime labeling is called a total neighborhood prime graph. In this paper, we examine total neighborhood prime labeling of trees such as (n, k, m) double star trees, spiders, caterpillars and firecrackers.

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Xue, Jie, Ruifang Liu, and Huicai Jia. "On the distance spectrum of trees." Filomat 30, no. 6 (2016): 1559–65. http://dx.doi.org/10.2298/fil1606559x.

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Let G be a connected graph with vertex set V(G) = {v1, v2,..., vng} and edge set E(G).D(G) = (dij)nxn is the distance matrix of G, where dij denotes the distance between vi and vj. Let ?1(D) ? ?2(D)?... ? ?n(D) be the distance spectrum of G. A graph G is said to be determined by its distance spectrum if any graph having the same distance spectrum as G is isomorphic to G. Trees can not be determined by its distance spectrum. Naturally, we prove that two kinds of special trees path Pn and double star S(a,b) are determined by their distance spectra in this paper.

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Dave, Siddharth, Ryan Clark, and Regina S. K. Lee. "RSOnet: An Image-Processing Framework for a Dual-Purpose Star Tracker as an Opportunistic Space Surveillance Sensor." Sensors 22, no. 15 (2022): 5688. http://dx.doi.org/10.3390/s22155688.

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A catalogue of over 22,000 objects in Earth’s orbit is currently maintained, and that number is expected to double within the next decade. Novel data collection regimes are needed to scale our ability to detect, track, classify and characterize resident space objects in a crowded low Earth orbit. This research presents RSOnet, an image-processing framework for space domain awareness using star trackers. Star trackers are cost-effective, flight proven, and require basic image processing to be used as an attitude-determination sensor. RSOnet is designed to augment the capabilities of a star tracker by becoming an opportunistic space-surveillance sensor. Our research demonstrates that star trackers are a feasible source for RSO detections in LEO by demonstrating the performance of RSOnet on real detections from a star-tracker-like imager in space. RSOnet convolutional-neural-network model architecture, graph-based multi-object classifier and characterization results are described in this paper.

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Lo Faro, Giovanni, and Antoinette Tripodi. "Bi-Squashing S2,2-Designs into (K4 − e)-Designs." Mathematics 12, no. 12 (2024): 1879. http://dx.doi.org/10.3390/math12121879.

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A double-star Sq1,q2 is the graph consisting of the union of two stars, K1,q1 and K1,q2, together with an edge joining their centers. The spectrum for Sq1,q2-designs, i.e., the set of all the n∈N such that an Sq1,q2-design of the order n exists, is well-known when q1=q2=2. In this article, S2,2-designs satisfying additional properties are investigated. We determine the spectrum for S2,2-designs that can be transformed into (K4−e)-designs by a double squash (bi-squash) passing through middle designs whose blocks are copies of a bull (the graph consisting of a triangle and two pendant edges). Here, the use of the difference method enables obtaining cyclic decompositions and determining the spectrum for cyclic S2,2-designs that can be purely bi-squashed into cyclic (K4−e)-designs (the middle bull designs are also cyclic).

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Khabyah, Ali Al, and Moin A. Ansari. "Exploring Geometrical Properties of Annihilator Intersection Graph of Commutative Rings." Axioms 14, no. 5 (2025): 336. https://doi.org/10.3390/axioms14050336.

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Let Λ denote a commutative ring with unity and D(Λ) denote a collection of all annihilating ideals from Λ. An annihilator intersection graph of Λ is represented by the notation AIG(Λ). This graph is not directed in nature, where the vertex set is represented by D(Λ)*. There is a connection in the form of an edge between two distinct vertices ς and ϱ in AIG(Λ) iff Ann(ςϱ)≠Ann(ς)∩Ann(ϱ). In this work, we begin by categorizing commutative rings Λ, which are finite in structure, so that AIG(Λ) forms a star graph/2-outerplanar graph, and we identify the inner vertex number of AIG(Λ). In addition, a classification of the finite rings where the genus of AIG(Λ) is 2, meaning AIG(Λ) is a double-toroidal graph, is also investigated. Further, we determine Λ, having a crosscap 1 of AIG(Λ), indicating that AIG(Λ) is a projective plane. Finally, we examine the domination number for the annihilator intersection graph and demonstrate that it is at maximum, two.

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Zhou, Zhong. "Super-stable Kneading Sequences with Double Cycles in 1D Bimodal Maps." Journal of Computing and Electronic Information Management 11, no. 3 (2023): 42–45. http://dx.doi.org/10.54097/jceim.v11i3.10.

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It is well known that a super-stable kneading sequence (SSKS) is an important concept, all SSKSs in bimodal maps forms joints in the corresponding symbolic dynamics, it decides the multiplication table of star products, which the n-tupling bifurcations to chaos can be investigated and Feigenbaum’s metric universalities can be measured and reconstructed, this SSKSs have form which are periodic with single cycle. However, in fact, the SSKSs in bimodal maps have another form with double cycles which are little mentioned and researched, they have the same position and significance as the single cycle SSKS. In the paper, we presented the number of admissible SSKSs with period-n and the joints graph on the parameter plane.

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Samadi, Babak, Morteza Alishahi, Iman Masoumi, and Doost Ali Mojdeh. "Restrained Italian domination in graphs." RAIRO - Operations Research 55, no. 2 (2021): 319–32. http://dx.doi.org/10.1051/ro/2021022.

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For a graph G = (V(G), E(G)), an Italian dominating function (ID function) f : V(G) → {0,1,2} has the property that for every vertex v ∈ V(G) with f(v) = 0, either v is adjacent to a vertex assigned 2 under f or v is adjacent to least two vertices assigned 1 under f. The weight of an ID function is ∑v∈V(G) f(v). The Italian domination number is the minimum weight taken over all ID functions of G. In this paper, we initiate the study of a variant of ID functions. A restrained Italian dominating function (RID function) f of G is an ID function of G for which the subgraph induced by {v ∈ V(G) | f(v) = 0} has no isolated vertices, and the restrained Italian domination number γrI (G) is the minimum weight taken over all RID functions of G. We first prove that the problem of computing this parameter is NP-hard, even when restricted to bipartite graphs and chordal graphs as well as planar graphs with maximum degree five. We prove that γrI(T) for a tree T of order n ≥ 3 different from the double star S2,2 can be bounded from below by (n + 3)/2. Moreover, all extremal trees for this lower bound are characterized in this paper. We also give some sharp bounds on this parameter for general graphs and give the characterizations of graphs G with small or large γrI (G).

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Journal articles on the topic 'Double star graph'

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