Artículos de revistas: "Connection graph"

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Ismail, Sumarno, Isran K. Hasan, Tesya Sigar, and Salmun K. Nasib. "RAINBOW CONNECTION NUMBER AND TOTAL RAINBOW CONNECTION NUMBER OF AMALGAMATION RESULTS DIAMOND GRAPH(〖Br〗_4) AND FAN GRAPH(F_3)." BAREKENG: Jurnal Ilmu Matematika dan Terapan 16, no. 1 (2022): 023–30. http://dx.doi.org/10.30598/barekengvol16iss1pp023-030.

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If be a graph and edge coloring of G is a function , rainbow connection number is the minimum-k coloration of the rainbow on the edge of graph G and denoted by rc(G). Rainbow connection numbers can be applied to the result of operations on some special graphs, such as diamond graphs and fan graphs. Graph operation is a method used to obtain a new graph by combining two graphs. This study performed amalgamation operations to obtain rainbow connection numbers and rainbow-total-connection numbers in diamond graphs ( ) and fan graphs ( ) or . Based on the research, it is obtained that the rainbow-connection number theorem on the amalgamation result of the diamond graph ( ) and fan graph ( is with . Furthermore, the theorem related to the total rainbow-connection number on the amalgamation result of the diamond graph( ) and the fan graph ( is obtained, namely with .

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Bustan, A. W., A. N. M. Salman, and P. E. Putri. "On the locating rainbow connection number of amalgamation of complete graphs." Journal of Physics: Conference Series 2543, no. 1 (2023): 012004. http://dx.doi.org/10.1088/1742-6596/2543/1/012004.

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Abstract Locating rainbow connection number determines the minimum number of colors connecting any two vertices of a graph with a rainbow vertex path and also verifies that the given colors produce a different rainbow code for each vertex. Locating rainbow connection number of graphs is a new mathematical concept, especially in graph theory, which combines the concepts of the rainbow vertex coloring and the partition dimension. In this paper, we determine the locating rainbow connection number of amalgamation of complete graphs.

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Alrowaili, Dalal Awadh, Faiz Farid, and Muhammad Javaid. "Gutman Connection Index of Graphs under Operations." Symmetry 15, no. 1 (2022): 21. http://dx.doi.org/10.3390/sym15010021.

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In the modern era, mathematical modeling consisting of graph theoretic parameters or invariants applied to solve the problems existing in various disciplines of physical sciences like computer sciences, physics, and chemistry. Topological indices (TIs) are one of the graph invariants which are frequently used to identify the different physicochemical and structural properties of molecular graphs. Wiener index is the first distance-based TI that is used to compute the boiling points of the paraffine. For a graph F, the recently developed Gutman Connection (GC) index is defined on all the unordered pairs of vertices as the sum of the multiplications of the connection numbers and the distance between them. In this note, the GC index of the operation-based symmetric networks called by first derived graph D1(F) (subdivision graph), second derived graph D2(F) (vertex-semitotal graph), third derived graph D3(F) (edge-semitotal graph) and fourth derived graph D4(F) (total graph) are computed in their general expressions consisting of various TIs of the parent graph F, where these operation-based symmetric graphs are obtained by applying the operations of subdivision, vertex semitotal, edge semitotal and the total on the graph F respectively.

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ZHANG, YINGYING, and XIAOYU ZHU. "Proper Vertex Connection and Graph Operations." Journal of Interconnection Networks 19, no. 02 (2019): 1950001. http://dx.doi.org/10.1142/s0219265919500014.

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A path in a vertex-colored graph is a vertex-proper path if any two internal adjacent vertices differ in color. A vertex-colored graph is proper vertex k-connected if any two vertices of the graph are connected by k disjoint vertex-proper paths of the graph. For a k-connected graph G, the proper vertex k-connection number of G, denoted by pvck(G), is defined as the smallest number of colors required to make G proper vertex k-connected. A vertex-colored graph is strong proper vertex-connected, if for any two vertices u, v of the graph, there exists a vertex-proper u-v geodesic. For a connected graph G, the strong proper vertex-connection number of G, denoted by spvc(G), is the smallest number of colors required to make G strong proper vertex-connected. In this paper, we study the proper vertex k-connection number and the strong proper vertex-connection number on the join of two graphs, the Cartesian, lexicographic, strong and direct product, and present exact values or upper bounds for these operations of graphs. Then we apply these results to some instances of Cartesian and lexicographic product networks.

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Farid, Faiz, Muhammad Javaid, and Ebenezer Bonyah. "Computing Connection Distance Index of Derived Graphs." Mathematical Problems in Engineering 2022 (July 18, 2022): 1–15. http://dx.doi.org/10.1155/2022/1439177.

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Distance based topological indices (TIs) play a vital role in the study of various structural and chemical aspects for the molecular graphs. The first distance-based TI is used to find the boiling point of paraffin. The connection distance (CD) index is a latest developed TI that is defined as the sum of all the products of distances between pair of vertices with the sum of their respective connection numbers . In this paper, we computed CD indices of the different derived graphs (subdivision graph S G , vertex-semitotal graph R G , edge-semitotal graph Q G and total graph T G obtained from the graph G under various operations of subdivision in the form of degree distance (DD) and CD indices of the basic graphs including some other algebraic expressions.

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Javaid, Muhammad, Muhammad Khubab Siddique, and Ebenezer Bonyah. "Computing Gutman Connection Index of Thorn Graphs." Journal of Mathematics 2021 (November 15, 2021): 1–13. http://dx.doi.org/10.1155/2021/2289514.

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Chemical structural formula can be represented by chemical graphs in which atoms are considered as vertices and bonds between them are considered as edges. A topological index is a real value that is numerically obtained from a chemical graph to predict its various physical and chemical properties. Thorn graphs are obtained by attaching pendant vertices to the different vertices of a graph under certain conditions. In this paper, a numerical relation between the Gutman connection (GC) index of a graph and its thorn graph is established. Moreover, the obtained result is also illustrated by computing the GC index for the particular families of the thorn graphs such as thorn paths, thorn rods, thorn stars, and thorn rings.

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Lihawa, Indrawati, Sumarno Ismail, Isran K. Hasan, Lailany Yahya, Salmun K. Nasib, and Nisky Imansyah Yahya. "Bilangan Terhubung Titik Pelangi pada Graf Hasil Operasi Korona Graf Prisma (P_(m,2)) dan Graf Lintasan (P_3)." Jambura Journal of Mathematics 4, no. 1 (2022): 145–51. http://dx.doi.org/10.34312/jjom.v4i1.11826.

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Rainbow vertex-connection number is the minimum k-coloring on the vertex graph G and is denoted by rvc(G). Besides, the rainbow-vertex connection number can be applied to some special graphs, such as prism graph and path graph. Graph operation is a method used to create a new graph by combining two graphs. Therefore, this research uses corona product operation to form rainbow-vertex connection number at the graph resulting from corona product operation of prism graph and path graph (Pm,2 P3) (P3 Pm,2). The results of this study obtain that the theorem of rainbow vertex-connection number at the graph resulting from corona product operation of prism graph and path graph (Pm,2 P3) (P3 Pm,2) for 3 = m = 7 are rvc (G) = 2m rvc (G) = 2.

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Yahya, Nisky Imansyah, Ainun Fatmawati, Nurwan Nurwan, and Salmun K. Nasib. "RAINBOW VERTEX-CONNECTION NUMBER ON COMB PRODUCT OPERATION OF CYCLE GRAPH (C_4) AND COMPLETE BIPARTITE GRAPH (K_(3,N))." BAREKENG: Jurnal Ilmu Matematika dan Terapan 17, no. 2 (2023): 0673–84. http://dx.doi.org/10.30598/barekengvol17iss2pp0673-0684.

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Rainbow vertex-connection number is the minimum colors assignment to the vertices of the graph, such that each vertex is connected by a path whose edges have distinct colors and is denoted by . The rainbow vertex connection number can be applied to graphs resulting from operations. One of the methods to create a new graph is to perform operations between two graphs. Thus, this research uses comb product operation to determine rainbow-vertex connection number resulting from comb product operation of cycle graph and complete bipartite graph & . The research finding obtains the theorem of rainbow vertex-connection number at the graph of for while the theorem of rainbow vertex-connection number at the graph of for for .

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Asif, Muhammad, Bartłomiej Kizielewicz, Atiq ur Rehman, Muhammad Hussain та Wojciech Sałabun. "Study of θϕ Networks via Zagreb Connection Indices". Symmetry 13, № 11 (2021): 1991. http://dx.doi.org/10.3390/sym13111991.

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Graph theory can be used to optimize interconnection network systems. The compatibility of such networks mainly depends on their topology. Topological indices may characterize the topology of such networks. In this work, we studied a symmetric network θϕ formed by ϕ time repetition of the process of joining θ copies of a selected graph Ω in such a way that corresponding vertices of Ω in all the copies are joined with each other by a new edge. The symmetry of θϕ is ensured by the involvement of complete graph Kθ in the construction process. The free hand to choose an initial graph Ω and formation of chemical graphs using θϕΩ enhance its importance as a family of graphs which covers all the pre-defined graphs, along with space for new graphs, possibly formed in this way. We used Zagreb connection indices for the characterization of θϕΩ. These indices have gained worth in the field of chemical graph theory in very small duration due to their predictive power for enthalpy, entropy, and acentric factor. These computations are mathematically novel and assist in topological characterization of θϕΩ to enable its emerging use.

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Ma, Yingbin, and Kairui Nie. "Rainbow Vertex Connection Numbers and Total Rainbow Connection Numbers of Middle and Total Graphs." Ars Combinatoria 157 (December 31, 2023): 45–52. http://dx.doi.org/10.61091/ars157-04.

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A vertex-colouring of a graph Γ is rainbow vertex connected if every pair of vertices ( u , v ) in Γ there is a u − v path whose internal vertices have different colours. The rainbow vertex connection number of a graph Γ , is the minimum number of colours needed to make Γ rainbow vertex connected, denoted by r v c ( Γ ) . Here, we study the rainbow vertex connection numbers of middle and total graphs. A total-colouring of a graph Γ is total rainbow connected if every pair of vertices ( u , v ) in Γ there is a u − v path whose edges and internal vertices have different colours. The total rainbow connection number of Γ , is the minimum number of colours required to colour the edges and vertices of Γ in order to make Γ total rainbow connected, denoted by t r c ( Γ ) . In this paper, we also research the total rainbow connection numbers of middle and total graphs.

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M. Trajković, Dragana, and Branislav Dimitrijević. "BOND GRAPH BICAUSALITY MODELING THE HYDRAULIC SYSTEM." KNOWLEDGE - International Journal 54, no. 3 (2022): 521–25. http://dx.doi.org/10.35120/kij5403521t.

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This papers represents one of the most widely challenging control problems. The most accepted blockdiagrams in automatic control used to describe processes have been replaced by control based on bond graphmodeling. The bond graph model one physical model hydraulic process is presented. The goal of the research is toobtain a model of process with and without knowledge of the mathematical model, which is used to obtain asimulation and prediction model. Due to the feedback effect of the liquid in the pipes (power and flow), a bicausalbond graph was used as the flow source. Bond graphs have a basic concept in their physics - energy that isexchanged through connectors 0 and 1 (ports). Effort e (force, voltage, pressure, etc.) and flow f (current, velocity,volume, etc.) are general physical quantities that are used to analyze the appropriate physical model and descriptionfor bond graph modeling and that very successfully. Bond-graph modeling is a powerful tool for modelingengineering systems, especially when physical domains are involved. Submodels graph can be reused, because linkgraph models are not causal. Connection graphs are labeled and directed graphs, in which vertices representsubmodels and arrows represent the ideal energy connection between power ports. Bond has a direction of strengthand a direction of causality. The assigned computational causality dictates which port variable will be computed asthe result (output) and accordingly, another port variable will be the cause (input). Graphs can be connected to partsof the block diagram, submodels of the connection graph can have power connections, signal inputs and signaloutputs as their interface elements. Aspects such as the physical domain of the connection (energy flow) can be usedto support the modeling process. The research in this work is on obtaining a fast and adequate physical model withgood knowledge of physical changes.

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Noda, Atsushi, Hideitsu Hino, Masami Tatsuno, Shotaro Akaho, and Noboru Murata. "Intrinsic Graph Structure Estimation Using Graph Laplacian." Neural Computation 26, no. 7 (2014): 1455–83. http://dx.doi.org/10.1162/neco_a_00603.

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A graph is a mathematical representation of a set of variables where some pairs of the variables are connected by edges. Common examples of graphs are railroads, the Internet, and neural networks. It is both theoretically and practically important to estimate the intensity of direct connections between variables. In this study, a problem of estimating the intrinsic graph structure from observed data is considered. The observed data in this study are a matrix with elements representing dependency between nodes in the graph. The dependency represents more than direct connections because it includes influences of various paths. For example, each element of the observed matrix represents a co-occurrence of events at two nodes or a correlation of variables corresponding to two nodes. In this setting, spurious correlations make the estimation of direct connection difficult. To alleviate this difficulty, a digraph Laplacian is used for characterizing a graph. A generative model of this observed matrix is proposed, and a parameter estimation algorithm for the model is also introduced. The notable advantage of the proposed method is its ability to deal with directed graphs, while conventional graph structure estimation methods such as covariance selections are applicable only to undirected graphs. The algorithm is experimentally shown to be able to identify the intrinsic graph structure.

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Ameliyah, Addinda Nur, and I. Ketut Budayasa. "BILANGAN KETERHUBUNGAN TITIK PELANGI BEBERAPA KELAS GRAF." MATHunesa: Jurnal Ilmiah Matematika 11, no. 3 (2023): 339–48. http://dx.doi.org/10.26740/mathunesa.v11n3.p339-348.

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A graph G is called a rainbow vertex connected if every two vertices G are connected by a rainbow path, that is, a path whose all the internal vertices are of a different color. The rainbow vertex connection number of graph G denoted by rvc(G) is the minimum number of colors used to color all vertices by G such that the graph G is connected to rainbow vertex. The rainbow vertex connection number in a graph will not be less than the diameter of the graph minus one. The rainbow vertex connection number discussed in this article for various classes of graphs include complete graph Kn, complete bipartite graph Km,n , wheel graph Wn , two-layer wheel graph Wn2, complete multipartite graph Kn1,n2,...,nt , path Pn, comb graph GSn, graph , graph , graph , graph . Keywords: graph, vertex coloring, rainbow vertex connection number.

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Zhou, Chao, and Yan Ping Liu. "Study on Parameter Transfer Structure of Generalized Modular Based Graph Theory." Advanced Materials Research 562-564 (August 2012): 1323–26. http://dx.doi.org/10.4028/www.scientific.net/amr.562-564.1323.

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For the purpose of reducing product structure levels and shorting transfer chain of parameter, in this paper the product structure levels are expressed with generalized modular. The concept of directed graph of parameter connection structure for generalized modular is proposed with the use of directed graph theory, generalized modular, sub-modular and part represented by vertex, the driven relations of parameter connection represented by directed edge, and the properties of directed graph of parameter connection structure for generalized modular are gained. The directed graph of parameter connection structure for generalized modular is divided into a number of sub-graphs according to the relations of product-level modular structure. And the horizontal edges of sub-graphs among vertexes are decomposed. Therefore, a standardized relation of parameter connection structure is established by given the decomposition algorithm and the mathematical description of parameters connection that are provide the theoretical basis for parameters connection analysis of variant design.

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Huntala, Melisa, Muhammad Rezky Friesta Payu, and Nisky Imansyah Yahya. "Total Rainbow Connection Number Of Shackle Product Of Antiprism Graph (〖AP〗_3)." Jurnal Matematika, Statistika dan Komputasi 20, no. 1 (2023): 1–9. http://dx.doi.org/10.20956/j.v20i1.24833.

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Function if is said to be k total rainbows in , for each pair of vertex there is a path called with each edge and each vertex on the path will have a different color. The total connection number is denoted by trc defined as the minimum number of colors needed to make graph to be total rainbow connected. Total rainbow connection numbers can also be applied to graphs that are the result of operations. The denoted shackle graph is a graph resulting from the denoted graph where t is number of copies of G. This research discusses rainbow connection numbers rc and total rainbow connection trc(G) using the shackle operation, where is the antiprism graph . Based on this research, rainbow connection numbers rc shack , and total rainbow connection trc shack for .

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Stojanović, Milica. "Properties of 3-Triangulations for p-Toroid." Athens Journal of Sciences 10, no. 1 (2023): 31–40. http://dx.doi.org/10.30958/ajs.10-1-2.

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In this paper, a method for constructing a toroid and its decomposition into convex pieces is considered. A graph of connection for 3-triangulable toroid is introduced in such a way that these pieces are represented by graph nodes. It is shown that connected, nonorientable graph can serve as a graph of connection for some of the toroids. The relationship between graphs that can be realized on surfaces of different genus and corresponding toroids is considered. Keywords: 3-triangulation of polyhedra, toroids, piecewise convex polyhedra, graph of connection

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Lestari, Dia, and I. Ketut Budayasa. "BILANGAN KETERHUBUNGAN PELANGI PADA PEWARNAAN-SISI GRAF." MATHunesa: Jurnal Ilmiah Matematika 8, no. 1 (2020): 25–34. http://dx.doi.org/10.26740/mathunesa.v8n1.p25-34.

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Let be a graph. An edge-coloring of is a function , where is a set of colors. Respect to a subgraph of is called a rainbow subgraph if all edges of get different colors. Graph is called rainbow connected if for every two distinct vertices of is joined by a rainbow path. The rainbow connection number of , denoted by , is the minimum number of colors needed in coloring all edges of such that is a rainbow connected. The main problem considered in this thesis is determining the rainbow connection number of graph. In this thesis, we determine the exact value of the rainbow connection number of some classes of graphs such as Cycles, Complete graph, and Tree. We also determining the lower bound and upper bound for the rainbow connection number of graph. Keywords: Rainbow Connection Number, Graph, Edge-Coloring on Graph.

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CSIKVÁRI, PÉTER, and ZOLTÁN LÓRÁNT NAGY. "The Density Turán Problem." Combinatorics, Probability and Computing 21, no. 4 (2012): 531–53. http://dx.doi.org/10.1017/s0963548312000016.

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LetHbe a graph onnvertices and let the blow-up graphG[H] be defined as follows. We replace each vertexviofHby a clusterAiand connect some pairs of vertices ofAiandAjif (vi,vj) is an edge of the graphH. As usual, we define the edge density betweenAiandAjasWe study the following problem. Given densities γijfor each edge (i,j) ∈E(H), one has to decide whether there exists a blow-up graphG[H], with edge densities at least γij, such that one cannot choose a vertex from each cluster, so that the obtained graph is isomorphic toH,i.e., noHappears as a transversal inG[H]. We calldcrit(H) the maximal value for which there exists a blow-up graphG[H] with edge densitiesd(Ai,Aj)=dcrit(H) ((vi,vj) ∈E(H)) not containingHin the above sense. Our main goal is to determine the critical edge density and to characterize the extremal graphs.First, in the case of treeTwe give an efficient algorithm to decide whether a given set of edge densities ensures the existence of a transversalTin the blow-up graph. Then we give general bounds ondcrit(H) in terms of the maximal degree. In connection with the extremal structure, the so-called star decomposition is proved to give the best construction forH-transversal-free blow-up graphs for several graph classes. Our approach applies algebraic graph-theoretical, combinatorial and probabilistic tools.

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Potočnik, Primož, Gabriel Verret, and Stephen Wilson. "Base graph–connection graph: Dissection and construction." Discrete Applied Mathematics 291 (March 2021): 116–28. http://dx.doi.org/10.1016/j.dam.2020.10.028.

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20

Li, Zhenzhen, and Baoyindureng Wu. "Maximum value of conflict-free vertex-connection number of graphs." Discrete Mathematics, Algorithms and Applications 10, no. 05 (2018): 1850059. http://dx.doi.org/10.1142/s1793830918500593.

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A path in a vertex-colored graph is called conflict-free if there is a color used on exactly one of its vertices. A vertex-colored graph is said to be conflict-free vertex-connected if any two vertices of the graph are connected by a conflict-free path. The conflict-free vertex-connection number, denoted by [Formula: see text], is defined as the smallest number of colors required to make [Formula: see text] conflict-free vertex-connected. Li et al. [Conflict-free vertex-connections of graphs, preprint (2017), arXiv:1705.07270v1[math.CO]] conjectured that for a connected graph [Formula: see text] of order [Formula: see text], [Formula: see text]. We confirm that the conjecture is true and poses two relevant conjectures.

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DEVROYE, LUC, and NICOLAS FRAIMAN. "The Random Connection Model on the Torus." Combinatorics, Probability and Computing 23, no. 5 (2014): 796–804. http://dx.doi.org/10.1017/s0963548313000631.

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We study the diameter of a family of random graphs on the torus that can be used to model wireless networks. In the random connection model two pointsxandyare connected with probabilityg(y−x), wheregis a given function. We prove that the diameter of the graph is bounded by a constant, which depends only on ‖g‖1, with high probability as the number of vertices in the graph tends to infinity.

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Septory, Brian Juned, Liliek Susilowaty, Dafik, V. Lokehsa, and G. Nagamani. "On the study of Rainbow Antimagic Connection Number of Corona Product of Graphs." European Journal of Pure and Applied Mathematics 16, no. 1 (2023): 271–85. http://dx.doi.org/10.29020/nybg.ejpam.v16i1.4520.

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Given that a graph G = (V, E). By an edge-antimagic vertex labeling of graph, we mean assigning labels on each vertex under the label function f : V → {1, 2, . . . , |V (G)|} such that the associated weight of an edge uv ∈ E(G), namely w(xy) = f(x) + f(y), has distinct weight. A path P in the vertex-labeled graph G is said to be a rainbow path if for every two edges xy, x′y ′ ∈ E(P) satisfies w(xy) ̸= w(x ′y ′ ). The function f is called a rainbow antimagic labeling of G if for every two vertices x and y of G, there exists a rainbow x − y path. When we assign each edge xy with the color of the edge weight w(xy), thus we say the graph G admits a rainbow antimagic coloring. The rainbow antimagic connection number of G, denoted by rac(G), is the smallest number of colors induced from all edge weight of antimagic labeling. In this paper, we will study the rac(G) of the corona product of graphs. By the corona product of graphs G and H, denoted by G ⊙ H, we mean a graph obtained by taking a copy of graph G and n copies of graph H, namely H1, H2, ..., Hn, then connecting vertex vi from the copy of graph G to every vertex on graph Hi , i = 1, 2, 3, . . . , n. In this paper, we show the exact value of the rainbow antimagic connection number of Tn ⊙ Sm where Tn ∈ {Pn, Sn, Sn,p, Fn,3}.

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Bustan, Ariestha W., A. N. M. Salman, Pritta E. Putri, and Zata Y. Awanis. "On the Locating Rainbow Connection Number of Trees and Regular Bipartite Graphs." Emerging Science Journal 7, no. 4 (2023): 1260–73. http://dx.doi.org/10.28991/esj-2023-07-04-016.

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Locating the rainbow connection number of graphs is a new mathematical concept that combines the concepts of the rainbow vertex coloring and the partition dimension. In this research, we determine the lower and upper bounds of the locating rainbow connection number of a graph and provide the characterization of graphs with the locating rainbow connection number equal to its upper and lower bounds to restrict the upper and lower bounds of the locating rainbow connection number of a graph. We also found the locating rainbow connection number of trees and regular bipartite graphs. The method used in this study is a deductive method that begins with a literature study related to relevant previous research concepts and results, making hypotheses, conducting proofs, and drawing conclusions. This research concludes that only path graphs with orders 2, 3, 4, and complete graphs have a locating rainbow connection number equal to 2 and the order of graph G, respectively. We also showed that the locating rainbow connection number of bipartite regular graphs is in the range of r-⌊n/4⌋+2 to n/2+1, and the locating rainbow connection number of a tree is determined based on the maximum number of pendants or the maximum number of internal vertices. Doi: 10.28991/ESJ-2023-07-04-016 Full Text: PDF

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Effendi, Bukti Ginting, Mutia Mawaddany, and Syafrizal Sy. "ON RAINBOW CONNECTION NUMBERS FOR LINE GRAPHS OF FAN GRAPH AND WINDMILL GRAPH." Far East Journal of Mathematical Sciences (FJMS) 103, no. 5 (2018): 931–39. http://dx.doi.org/10.17654/ms103050931.

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Ma, Yingbin, Xiaoxue Zhang, and Yanfeng Xue. "Graphs with Strong Proper Connection Numbers and Large Cliques." Axioms 12, no. 4 (2023): 353. http://dx.doi.org/10.3390/axioms12040353.

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In this paper, we mainly investigate graphs with a small (strong) proper connection number and a large clique number. First, we discuss the (strong) proper connection number of a graph G of order n and ω(G)=n−i for 1⩽i⩽3. Next, we investigate the rainbow connection number of a graph G of order n, diam(G)≥3 and ω(G)=n−i for 2⩽i⩽3.

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Nguyen Thi Thuy, Anh, and Duyen Le Thi. "A NOTE ON GENERALIZED RAINBOW CONNECTION OF CONNECTED GRAPHS AND THEIR NUMBER OF EDGES." Journal of Science Natural Science 66, no. 3 (2021): 3–7. http://dx.doi.org/10.18173/2354-1059.2021-0041.

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Let l ≥ 1, k ≥ 1 be two integers. Given an edge-coloured connected graph G. A path P in the graph G is called l-rainbow path if each subpath of length at most l + 1 is rainbow. The graph G is called (k, l)-rainbow connected if any two vertices in G are connected by at least k pairwise internally vertex-disjoint l-rainbow paths. The smallest number of colours needed in order to make G (k, l)-rainbow connected is called the (k, l)-rainbow connection number of G and denoted by rck,l(G). In this paper, we first focus to improve the upper bound of the (1, l)-rainbow connection number depending on the size of connected graphs. Using this result, we characterize all connected graphs having the large (1, 2)-rainbow connection number. Moreover, we also determine the (1, l)-rainbow connection number in a connected graph G containing a sequence of cut-edges.

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Marhelina, Sally. "RAINBOW CONNECTION PADA GRAF k -CONNECTED UNTUK k = 1 ATAU 2." Jurnal Matematika UNAND 2, no. 1 (2013): 78. http://dx.doi.org/10.25077/jmu.2.1.78-84.2013.

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An edge-colored graph G is rainbow connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection number of aconnected graph G, denoted by rc(G) is the smallest number of colors needed such thatG is rainbow connected. In this paper, we will proved again that rc(G) ≤ 3(n + 1)/5 forall 3-connected graphs, and rc(G) ≤ 2n/3 for all 2-connected graphs.

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Wang, Yexin, Zhi Yang, Junqi Liu, Wentao Zhang, and Bin Cui. "Scapin: Scalable Graph Structure Perturbation by Augmented Influence Maximization." Proceedings of the ACM on Management of Data 1, no. 2 (2023): 1–21. http://dx.doi.org/10.1145/3589291.

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Generating data perturbations to graphs has become a useful tool for analyzing the robustness of Graph Neural Networks (GNNs). However, existing model-driven methodologies can be prohibitively expensive to apply in large graphs, which hinders the understanding of GNN robustness at scale. In this paper, we present Scapin, a data-driven methodology that opens up a new perspective by connecting graph structure perturbation for GNNs with augmented influence maximization-to either facilitate desirable spreads or curtail undesirable ones by adding or deleting a small set of edges. This connection not only allows us to perform data perturbation on GNNs with computation scalability but also provides nice interpretations. To transform such connections into efficient perturbation approaches for the new GNN setting, Scapin introduces a novel edge influence model, decomposed influence maximization objectives, and a principled algorithm for edge addition by exploiting submodularity of the objectives. Empirical studies demonstrate that Scapin can give orders of magnitude improvement over state-of-art methods in terms of runtime and memory efficiency, with comparable or even better performance.

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29

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

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A graph is a connection of objects. These objects are often known as vertices or nodes and the connection or relation in these nodes are called arcs or edges. There are certain rules to allocate values to these vertices and edges. This allocation of values to vertices or edges is called graph labeling. Labeling is prime cordial if vertices have allocated values from 1 to the order of graph and edges have allocated values 0 or 1 on a certain pattern. That is, an edge has an allocated value of 0 if the incident vertices have a greatest common divisor (gcd) greater than 1. An edge has an allocated value of 1 if the incident vertices have a greatest common divisor equal to 1. The number of edges labeled with 0 or 1 are equal in numbers or, at most, have a difference of 1. In this paper, our aim is to investigate the prime cordial labeling of rotationally symmetric graphs obtained from a generalized Petersen graph P(n,k) under duplication operation, and we have proved that the resulting symmetric graphs are prime cordial. Moreover, we have also proved that when we glow a Petersen graph with some path graphs, then again, the resulting graph is a prime cordial graph.

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30

Gologranc, Tanja, Gašper Mekiš, and Iztok Peterin. "Rainbow Connection and Graph Products." Graphs and Combinatorics 30, no. 3 (2013): 591–607. http://dx.doi.org/10.1007/s00373-013-1295-y.

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31

Cadavid, Paula, Mary Luz Rodiño Montoya, and Pablo M. Rodriguez. "The connection between evolution algebras, random walks and graphs." Journal of Algebra and Its Applications 19, no. 02 (2019): 2050023. http://dx.doi.org/10.1142/s0219498820500231.

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Evolution algebras are a new type of non-associative algebras which are inspired from biological phenomena. A special class of such algebras, called Markov evolution algebras, is strongly related to the theory of discrete time Markov chains. The winning of this relation is that many results coming from Probability Theory may be stated in the context of Abstract Algebra. In this paper, we explore the connection between evolution algebras, random walks and graphs. More precisely, we study the relationships between the evolution algebra induced by a random walk on a graph and the evolution algebra determined by the same graph. Given that any Markov chain may be seen as a random walk on a graph, we believe that our results may add a new landscape in the study of Markov evolution algebras.

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32

Naz, Kiran, Sarfraz Ahmad, and Eihab Bashier. "On Computing Techniques for Sombor Index of Some Graphs." Mathematical Problems in Engineering 2022 (October 10, 2022): 1–13. http://dx.doi.org/10.1155/2022/1329653.

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In all types of topological indicators, degree-based indicators play a major role in chemical graph theory. The topological index is a fixed numeric value associated with graph isomerism. Firstly, in 1972, the concept of degree-based index was developed by Gutman and Trinajstic. These degree-based indices are divided into two ways, namely, degree and connection number. These degree-based graph indices are positive-valued for non-regular graphs and zero for regular graphs. In this article, we discussed the degree-based Sombor, reduced Sombor, and average Sombor indices for wheel graph, gear graph, helm graph, flower graph, sunflower graph, and lobster graph.

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33

Mahasinghe, A. C., K. K. W. H. Erandi, and S. S. N. Perera. "Optimizing Wiener and Randić Indices of Graphs." Advances in Operations Research 2020 (September 26, 2020): 1–10. http://dx.doi.org/10.1155/2020/3139867.

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Wiener and Randić indices have long been studied in chemical graph theory as connection strength measures of graphs. Later, these indices were used in different fields such as network analysis. We consider two optimization problems related to these indices, with potential applications to network theory, in particular to epidemiological networks. Given a connected graph and a fixed total edge weight, we investigate how individual weights must be assigned to edges, minimizing the connection strength of the graph. In order to measure the connection strength, we use the weighted Wiener index and a modified version of the ordinary Randić index. Wiener index optimization is linear, while Randić index optimization turns out to be both nonlinear and nonconvex. Hence, we adopt the technique of separable programming to generate solutions. We present our experimental results by applying relevant algorithms to several graphs.

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34

Ma, Yingbin, and Hui Zhang. "Some results on the total proper k-connection number." Open Mathematics 20, no. 1 (2022): 195–209. http://dx.doi.org/10.1515/math-2022-0025.

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Abstract In this paper, we first investigate the total proper connection number of a graph G G according to some constraints of G ¯ \overline{G} . Next, we investigate the total proper connection numbers of graph G G with large clique number ω ( G ) = n − s \omega \left(G)=n-s for 1 ≤ s ≤ 3 1\le s\le 3 . Finally, we determine the total proper k k -connection numbers of circular ladders, Möbius ladders and all small cubic graphs of order 8 or less.

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35

Basavaraju, Manu, L. Sunil Chandran, Deepak Rajendraprasad, and Arunselvan Ramaswamy. "Rainbow Connection Number of Graph Power and Graph Products." Graphs and Combinatorics 30, no. 6 (2013): 1363–82. http://dx.doi.org/10.1007/s00373-013-1355-3.

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36

Robert, Y., and M. Tchuente. "Connection-graph and iteration-graph of monotone boolean functions." Discrete Applied Mathematics 11, no. 3 (1985): 245–53. http://dx.doi.org/10.1016/0166-218x(85)90076-9.

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37

Razumovsky, P. V., and M. B. Abrosimov. "THE MINIMAL VERTEX EXTENSIONS FOR COLORED COMPLETE GRAPHS." Bulletin of the South Ural State University series "Mathematics. Mechanics. Physics" 13, no. 4 (2021): 77–89. http://dx.doi.org/10.14529/mmph210409.

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The article proposes the results of the search for minimal vertex extensions of undirected colored complete graphs. The research topic is related to the modelling of full fault tolerant technical systems with a different type of their objects in the terminology of graph theory. Let a technical system be Σ, then there is a graph G(Σ), which vertices reflects system’s objects and edges reflects connections between these objects. Type of each object reflected in a mapping of some color from F = {1,2…,i} to the corresponding vertex. System’s Σ vertex extension is a graph G(Σ) which contains additional vertices. System reflected by graph G(Σ) can work even if there are k faults of its objects. Complete graph is a graph where each two vertices have an edge between them. Complete graphs have no edge extensions because there is no way to add additional edge to the graph with a maximum number of edges. In other words, the system reflected by some complete graph cannot be able to resist connection faults. Therefore the article research is focused on vertex extensions only. There is a description of vertex extensions existence condition for those colored complete graphs. This paper considers generating schemes for such minimal vertex extensions along with formulas, which allows to calculate number of additional edges to have an ability to construct minimal vertex extension.

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38

Dyke, Frances Van. "Activities for Students: Using Graphs to Introduce Functions." Mathematics Teacher 96, no. 2 (2003): 126–37. http://dx.doi.org/10.5951/mt.96.2.0126.

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HE PAST DECADE HAS SEEN A SHIFT TOWARD FUNCTIONS as a central theme in beginning algebra. The advent of graphing calculators has meant that the graphical representation of functions is accessible and can be used in a meaningful way. Yet evidence indicates that students ignore graphs and resort to complicated algebraic expressions rather than read information from a graph. In “Understanding Connections between Equations and Graphs,” Eric Knuth (2000) reported on a study that he conducted with 178 students from a suburban high school. He concluded that students may be missing the basic “Cartesian connection” and that they do not recognize that a graph and an equation are two representations for the same set of points.

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39

Yao, Yuhang, and Carlee Joe-Wong. "Interpretable Clustering on Dynamic Graphs with Recurrent Graph Neural Networks." Proceedings of the AAAI Conference on Artificial Intelligence 35, no. 5 (2021): 4608–16. http://dx.doi.org/10.1609/aaai.v35i5.16590.

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We study the problem of clustering nodes in a dynamic graph, where the connections between nodes and nodes' cluster memberships may change over time, e.g., due to community migration. We first propose a dynamic stochastic block model that captures these changes, and a simple decay-based clustering algorithm that clusters nodes based on weighted connections between them, where the weight decreases at a fixed rate over time. This decay rate can then be interpreted as signifying the importance of including historical connection information in the clustering. However, the optimal decay rate may differ for clusters with different rates of turnover. We characterize the optimal decay rate for each cluster and propose a clustering method that achieves almost exact recovery of the true clusters. We then demonstrate the efficacy of our clustering algorithm with optimized decay rates on simulated graph data. Recurrent neural networks (RNNs), a popular algorithm for sequence learning, use a similar decay-based method, and we use this insight to propose two new RNN-GCN (graph convolutional network) architectures for semi-supervised graph clustering. We finally demonstrate that the proposed architectures perform well on real data compared to state-of-the-art graph clustering algorithms.

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40

Mao, Yaping, Zhao Wang, Fengnan Yanling, and Chengfu Ye. "Monochromatic connectivity and graph products." Discrete Mathematics, Algorithms and Applications 08, no. 01 (2016): 1650011. http://dx.doi.org/10.1142/s1793830916500117.

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The concept of monochromatic connectivity was introduced by Caro and Yuster. A path in an edge-colored graph is called a monochromatic path if all the edges on the path are colored the same. An edge-coloring of [Formula: see text] is a monochromatic connection coloring ([Formula: see text]-coloring, for short) if there is a monochromatic path joining any two vertices in [Formula: see text]. The monochromatic connection number, denoted by [Formula: see text], is defined to be the maximum number of colors used in an [Formula: see text]-coloring of a graph [Formula: see text]. In this paper, we study the monochromatic connection number on the lexicographical, strong, Cartesian and direct products and present several upper and lower bounds for these products of graphs.

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41

Guo, Zhijiang, Yan Zhang, Zhiyang Teng, and Wei Lu. "Densely Connected Graph Convolutional Networks for Graph-to-Sequence Learning." Transactions of the Association for Computational Linguistics 7 (November 2019): 297–312. http://dx.doi.org/10.1162/tacl_a_00269.

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We focus on graph-to-sequence learning, which can be framed as transducing graph structures to sequences for text generation. To capture structural information associated with graphs, we investigate the problem of encoding graphs using graph convolutional networks (GCNs). Unlike various existing approaches where shallow architectures were used for capturing local structural information only, we introduce a dense connection strategy, proposing a novel Densely Connected Graph Convolutional Network (DCGCN). Such a deep architecture is able to integrate both local and non-local features to learn a better structural representation of a graph. Our model outperforms the state-of-the-art neural models significantly on AMR-to-text generation and syntax-based neural machine translation.

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42

Wang, Wenhu, Asma Nisar, Asfand Fahad, Muhammad Imran Qureshi, and Abdu Alameri. "Modified Zagreb Connection Indices for Benes Network and Related Classes." Journal of Mathematics 2022 (March 28, 2022): 1–8. http://dx.doi.org/10.1155/2022/8547332.

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The study of networks such as Butterfly networks, Benes networks, interconnection networks, David-derived networks through graph theoretical parameters is among the modern trends in the area of graph theory. Among these graph theoretical tools, the topological Indices TIs have been frequently used as graph invariants. TIs are also the essential tools for quantitative structure activity relationship (QSAR) as well as quantity structure property relationships (QSPR). TIs depend on different parameters, such as degree and distance of vertices in graphs. The current work is devoted to the derivation of 2-distance based TIs, known as, modified first Zagreb connection index ZC 1 ∗ and first Zagreb connection index ZC 1 for r − dimensional Benes network and some classes generated from Benes network. The horizontal cylindrical Benes network HCB r , vertical cylindrical Benes network VCB r , and toroidal Benes network TB r are the three classes generated by identifying the vertices of the first row with the last row, the first column with the last column of the Benes network. The obtained results are also analyzed through graphical tools.

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43

Yuniarti, Ervie Yuniarti Astika Mustafaputri, Budi Nurwahyu, and Jusmawati Massalesse. "Rainbow Connection Number of Double Quadrilateral Snake Graph." Jurnal Matematika, Statistika dan Komputasi 20, no. 1 (2023): 268–80. http://dx.doi.org/10.20956/j.v20i1.28141.

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Let graph G = be a non trivial connected graph. A graph G with edge coloring is called a rainbow connection, if for every pair of vertices on a path has a different color. The rainbow connection number denoted by is the minimum color needed to make graph G rainbow connection. In this study, we will determine the rainbow connection number of double quadrilateral snake graph and alternate double quadrilateral snake graph. The research results show that while if and if

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44

Farkhondeh, Masoumeh, Mohammad Habibi, Doost Ali Mojdeh, and Yongsheng Rao. "Some Bicyclic Graphs Having 2 as Their Laplacian Eigenvalues." Mathematics 7, no. 12 (2019): 1233. http://dx.doi.org/10.3390/math7121233.

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If G is a graph, its Laplacian is the difference between the diagonal matrix of its vertex degrees and its adjacency matrix. A one-edge connection of two graphs G 1 and G 2 is a graph G = G 1 ⊙ u v G 2 with V ( G ) = V ( G 1 ) ∪ V ( G 2 ) and E ( G ) = E ( G 1 ) ∪ E ( G 2 ) ∪ { e = u v } where u ∈ V ( G 1 ) and v ∈ V ( G 2 ) . In this paper, we study some structural conditions ensuring the presence of 2 in the Laplacian spectrum of bicyclic graphs of type G 1 ⊙ u v G 2 . We also provide a condition under which a bicyclic graph with a perfect matching has a Laplacian eigenvalue 2. Moreover, we characterize the broken sun graphs and the one-edge connection of two broken sun graphs by their Laplacian eigenvalue 2.

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45

Ivanko, Evgeny, and Mikhail Chernoskutov. "The Random Plots Graph Generation Model for Studying Systems with Unknown Connection Structures." Entropy 24, no. 2 (2022): 297. http://dx.doi.org/10.3390/e24020297.

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We consider the problem of modeling complex systems where little or nothing is known about the structure of the connections between the elements. In particular, when such systems are to be modeled by graphs, it is unclear what vertex degree distributions these graphs should have. We propose that, instead of attempting to guess the appropriate degree distribution for a poorly understood system, one should model the system via a set of sample graphs whose degree distributions cover a representative range of possibilities and account for a variety of possible connection structures. To construct such a representative set of graphs, we propose a new random graph generator, Random Plots, in which we (1) generate a diversified set of vertex degree distributions and (2) target a graph generator at each of the constructed distributions, one-by-one, to obtain the ensemble of graphs. To assess the diversity of the resulting ensembles, we (1) substantialize the vague notion of diversity in a graph ensemble as the diversity of the numeral characteristics of the graphs within this ensemble and (2) compare such formalized diversity for the proposed model with that of three other common models (Erdos–Rényi–Gilbert (ERG), scale-free, and small-world). Computational experiments show that, in most cases, our approach produces more diverse sets of graphs compared with the three other models, including the entropy-maximizing ERG. The corresponding Python code is available at GitHub.

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46

Bessouf, Ouahiba, and Abdelkader Khelladi. "New concept of connection in bidirected graphs." RAIRO - Operations Research 52, no. 2 (2018): 351–57. http://dx.doi.org/10.1051/ro/2017053.

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In bidirected graph an edge has a direction at each end. We introduce a new definition of connection in a bidirected graph. We prove some properties of this definition and we establish a relationship to connection and imbalance in the corresponding signed graph. The main result gives a sufficient condition for a signed graph to have a Biconnected biorientation.

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47

Joedo, J. C., Dafik, A. I. Kristiana, I. H. Agustin, and R. Nisviasari. "On the rainbow antimagic coloring of vertex amalgamation of graphs." Journal of Physics: Conference Series 2157, no. 1 (2022): 012014. http://dx.doi.org/10.1088/1742-6596/2157/1/012014.

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Abstract The purpose of this study is to develop rainbow antimagic coloring. This study is a combination of two notions, namely antimagic and rainbow concept. If every vertex of graph G is labeled with the antimagic labels and then edge weight of antimagic labels are used to assign a rainbow coloring. The minimum number of colors for a rainbow path to exist with the condition satisfying the edge weights w(x) ≠ w(y) for any two vertices x and y is the definition of the rainbow antimagic connection number rac(G). In this study, we use connected graphs and simple graphs in obtaining the rainbow antimagic connection number. This paper will explain the rainbow antimagic coloring on some graphs and get their formula of the rainbow antimagic connection number. We have obtained rac(G) where G is vertex amalgamation of graphs, namely path, star, broom, paw, fan, and triangular book graph.

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48

Wang, Jing, Songhe Feng, Gengyu Lyu, and Jiazheng Yuan. "SURER: Structure-Adaptive Unified Graph Neural Network for Multi-View Clustering." Proceedings of the AAAI Conference on Artificial Intelligence 38, no. 14 (2024): 15520–27. http://dx.doi.org/10.1609/aaai.v38i14.29478.

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Deep Multi-view Graph Clustering (DMGC) aims to partition instances into different groups using the graph information extracted from multi-view data. The mainstream framework of DMGC methods applies graph neural networks to embed structure information into the view-specific representations and fuse them for the consensus representation. However, on one hand, we find that the graph learned in advance is not ideal for clustering as it is constructed by original multi-view data and localized connecting. On the other hand, most existing methods learn the consensus representation in a late fusion manner, which fails to propagate the structure relations across multiple views. Inspired by the observations, we propose a Structure-adaptive Unified gRaph nEural network for multi-view clusteRing (SURER), which can jointly learn a heterogeneous multi-view unified graph and robust graph neural networks for multi-view clustering. Specifically, we first design a graph structure learning module to refine the original view-specific attribute graphs, which removes false edges and discovers the potential connection. According to the view-specific refined attribute graphs, we integrate them into a unified heterogeneous graph by linking the representations of the same sample from different views. Furthermore, we use the unified heterogeneous graph as the input of the graph neural network to learn the consensus representation for each instance, effectively integrating complementary information from various views. Extensive experiments on diverse datasets demonstrate the superior effectiveness of our method compared to other state-of-the-art approaches.

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49

Mooduto, Randi, Lailany Yahya, and Nisky Imansyah Yahya. "Total Rainbow Connection Number of Corona Product of Book Graph(Bn) and Pencil Graf(Pcm)." Sainsmat : Jurnal Ilmiah Ilmu Pengetahuan Alam 12, no. 2 (2023): 153. http://dx.doi.org/10.35580/sainsmat122423112023.

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Let G be a simple and finite graph. Rainbow connection and total rainbow connection c are set c : G → {1,2,. . . , k} where k is the minimal color on graph G. A rainbow connection number(rc) is a pattern by giving different colors to the connection edges (E(G)) so that a rainbow path is formed. The total rainbow connection number (trc) is a payment pattern by giving color to vertices (V(G)) and edges (E(G)) in graph G so that a total rainbow path is formed. This article discusses rainbow connection numbers (rc) and total rainbow connection numbers (trc) in the corona graph of book graph (Bn) and pencil graph (Pcm). The results obtained are rc(Bn ⨀ Pcm) = 2n+3 and trc(Bn ⨀ Pcm) = 4n+5, 3 ≤ n ≤ 5.

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50

Mao, Yaping, Fengnan Yanling, Zhao Wang, and Chengfu Ye. "Rainbow vertex-connection and graph products." International Journal of Computer Mathematics 93, no. 7 (2015): 1078–92. http://dx.doi.org/10.1080/00207160.2015.1047356.

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