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Artykuły w czasopismach na temat „Trees (graph theory)”

Autor: Grafiati

Data publikacji: 4 czerwca 2021

Data aktualizacji: 25 lipca 2024

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1

McKee, Terry A. "Subgraph trees in graph theory." Discrete Mathematics 270, no. 1-3 (August 2003): 3–12. http://dx.doi.org/10.1016/s0012-365x(03)00161-4.

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Thenge, J. D., B. Surendranath Reddy, and Rupali S. Jain. "Contribution to Soft Graph and Soft Tree." New Mathematics and Natural Computation 15, no. 01 (December 25, 2018): 129–43. http://dx.doi.org/10.1142/s179300571950008x.

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Soft set theory introduced by D. Molodstov is a new theory which deals with uncertainty. Connected graphs can be represented by using soft sets called soft graphs. In the present paper, we introduce the tabular representation of soft graph and define radius, diameter, center and degree of soft graph. We also define union, product of soft graphs and soft trees. We then derive some properties of radius, degree of vertex in soft graph and soft trees.

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3

Козневски and E. Kozniewski. "Roof Skeletons and Graph Theory Trees." Geometry & Graphics 4, no. 1 (March 17, 2016): 12–20. http://dx.doi.org/10.12737/18054.

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The problem of constructing roofs of many papers [1; 6; 7; 9; etc.]. In this case, some studies suggested to use a computer and special programs [10; 13; 16; etc.]. The geometry of the efficient design of roofs is actual scientific direction, so in the educational process of architectural and building trades made corresponding innovations [2; 8; 14; 15; 17–20; etc.]. The roofs considered in this article are defined as special geometric polyhedral surfaces, on the basis of two assumptions: (1) all eaves of a roof form a planar (simply-connected or

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4

Janson, Svante. "Random trees in a graph and trees in a random graph." Mathematical Proceedings of the Cambridge Philosophical Society 100, no. 2 (September 1986): 319–30. http://dx.doi.org/10.1017/s0305004100066111.

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This paper treats two related sets of problems in the theory of random graphs. In Sections 2 and 3 we study random spanning subtrees of a complete graph (or, equivalently, random labelled trees). It is shown that the number of common edges of two such random trees asymptotically has a Poisson distribution with expectation 2. Similar results are obtained for the number of edges in the intersection or union of more than two random trees.

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5

Markhabatov, Nurlan. "Approximations of Acyclic Graphs." Bulletin of Irkutsk State University. Series Mathematics 40 (2022): 104–11. http://dx.doi.org/10.26516/1997-7670.2022.40.104.

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In this paper, approximations of acyclic graphs are studied. It is proved that any theory of an acyclic graph (tree) of finite diameter is pseudofinite with respect to acyclic graphs (trees), that is, any such theory is approximated by theories of finite structures (acyclic graphs, trees). It is also proved that an acyclic graph of infinite diameter with infinite number of rays is pseudofinite.

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6

Zeen El Deen, Mohamed R. "Enumeration of spanning trees in prisms of some graphs." Proyecciones (Antofagasta) 42, no. 2 (December 1, 2023): 339–91. http://dx.doi.org/10.22199/issn.0717-6279-4664.

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In graph theory, a prism over a graph G is the cartesian product of the graph G with P₂. The purpose of this work is to investigate the complexity of the prisms of some path and cycle-related graphs. In particular, we obtain simpler and more explicit formulas for the complexity of a special class of prisms of path-related graphs: fan graph, ladder graph, the composition Pn[P₂] graph, and book graph. Moreover, we obtain straightforward formulas for the complexity of a special class of prisms of cycle-related graphs: wheel graph, gear graph, prism graph, n−crossed prism graph, mirror graph M(Cn)

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7

JOHANNSEN, DANIEL, MICHAEL KRIVELEVICH, and WOJCIECH SAMOTIJ. "Expanders Are Universal for the Class of All Spanning Trees." Combinatorics, Probability and Computing 22, no. 2 (January 3, 2013): 253–81. http://dx.doi.org/10.1017/s0963548312000533.

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A graph is calleduniversalfor a given graph class(or, equivalently,-universal) if it contains a copy of every graph inas a subgraph. The construction of sparse universal graphs for various classeshas received a considerable amount of attention. There is particular interest in tight-universal graphs, that is, graphs whose number of vertices is equal to the largest number of vertices in a graph from. Arguably, the most studied case is that whenis some class of trees. In this work, we are interested in(n,Δ), the class of alln-vertex trees with maximum degree at most Δ. We show that everyn-vertex

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8

Raczek, Joanna, and Rita Zuazua. "Progress on Roman and Weakly Connected Roman Graphs." Mathematics 9, no. 16 (August 5, 2021): 1846. http://dx.doi.org/10.3390/math9161846.

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A graph G for which γR(G)=2γ(G) is the Roman graph, and if γRwc(G)=2γwc(G), then G is the weakly connected Roman graph. In this paper, we show that the decision problem of whether a bipartite graph is Roman is a co-NP-hard problem. Next, we prove similar results for weakly connected Roman graphs. We also study Roman trees improving the result of M.A. Henning’s A characterization of Roman trees, Discuss. Math. Graph Theory 22 (2002). Moreover, we give a characterization of weakly connected Roman trees.

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9

Finbow, Stephen, та Christopher M. van Bommel. "γ-Graphs of Trees". Algorithms 12, № 8 (30 липня 2019): 153. http://dx.doi.org/10.3390/a12080153.

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For a graph     G = ( V , E )    , the    γ   -graph of G, denoted     G ( γ ) = ( V ( γ ) , E ( γ ) )    , is the graph whose vertex set is the collection of minimum dominating sets, or    γ   -sets of G, and two    γ   -sets are adjacent in    

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10

NATH, MILAN, and SOMNATH PAUL. "GRAPH TRANSFORMATION AND DISTANCE SPECTRAL RADIUS." Discrete Mathematics, Algorithms and Applications 05, no. 03 (September 2013): 1350014. http://dx.doi.org/10.1142/s1793830913500146.

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Trees are very common in the theory and applications of combinatorics. In this paper, we consider graphs whose underlying structure is a tree and study the behavior of the distance spectral radius under a graph transformation. As an application, we find the corona tree that maximizes the distance spectral radius among all corona trees with a fixed maximum degree. We also find the graph with minimal (maximal) distance spectral radius among all corona trees. Finally, we determine the graph with minimal distance spectral radius in a special class of corona trees.

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11

El-Mesady, A., Omar Bazighifan, and S. S. Askar. "A Novel Approach for Cyclic Decompositions of Balanced Complete Bipartite Graphs into Infinite Graph Classes." Journal of Function Spaces 2022 (May 4, 2022): 1–12. http://dx.doi.org/10.1155/2022/9308708.

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Graph theory is considered an attractive field for finding the proof techniques in discrete mathematics. The results of graph theory have applications in many areas of social, computing, and natural sciences. Graph labelings and decompositions have received much attention in the literature. Several types of graph labeling were proposed for solving the problem of decomposing different graph classes. In the present paper, we propose a technique for labeling the vertices of a bipartite graph G with n edges, called orthogonal labeling, to yield cyclic decompositions of balanced complete bipartite

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12

Simonet, Geneviève, and Anne Berry. "Properties and Recognition of Atom Graphs." Algorithms 15, no. 8 (August 19, 2022): 294. http://dx.doi.org/10.3390/a15080294.

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The atom graph of a connected graph is a graph whose vertices are the atoms obtained by clique minimal separator decomposition of this graph, and whose edges are the edges of all its atom trees. A graph G is an atom graph if there is a graph whose atom graph is isomorphic to G. We study the class of atom graphs, which is also the class of atom graphs of chordal graphs, and the associated recognition problem. We prove that each atom graph is a perfect graph and give a characterization of atom graphs in terms of a spanning tree, inspired by the characterization of clique graphs of chordal graphs

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13

Hosamani, S. M., V. B. Awati, and R. M. Honmore. "On graphs with equal dominating and c-dominating energy." Applied Mathematics and Nonlinear Sciences 4, no. 2 (December 24, 2019): 503–12. http://dx.doi.org/10.2478/amns.2019.2.00047.

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AbstractGraph energy and domination in graphs are most studied areas of graph theory. In this paper we try to connect these two areas of graph theory by introducing c-dominating energy of a graph G. First, we show the chemical applications of c-dominating energy with the help of well known statistical tools. Next, we obtain mathematical properties of c-dominating energy. Finally, we characterize trees, unicyclic graphs, cubic and block graphs with equal dominating and c-dominating energy.

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14

Zuki, Wan Nor Nabila Nadia Wan, Zhibin Du, Muhammad Kamran Jamil, and Roslan Hasni. "Extremal Trees with Respect to the Difference between Atom-Bond Connectivity Index and Randić Index." Symmetry 12, no. 10 (September 25, 2020): 1591. http://dx.doi.org/10.3390/sym12101591.

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Let G be a simple, connected and undirected graph. The atom-bond connectivity index (ABC(G)) and Randić index (R(G)) are the two most well known topological indices. Recently, Ali and Du (2017) introduced the difference between atom-bond connectivity and Randić indices, denoted as ABC−R index. In this paper, we determine the fourth, the fifth and the sixth maximum chemical trees values of ABC−R for chemical trees, and characterize the corresponding extremal graphs. We also obtain an upper bound for ABC−R index of such trees with given number of pendant vertices. The role of symmetry has great

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15

KIYOMI, MASASHI, TOSHIKI SAITOH, and RYUHEI UEHARA. "BIPARTITE PERMUTATION GRAPHS ARE RECONSTRUCTIBLE." Discrete Mathematics, Algorithms and Applications 04, no. 03 (August 6, 2012): 1250039. http://dx.doi.org/10.1142/s1793830912500395.

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The graph reconstruction conjecture is a long-standing open problem in graph theory. The conjecture has been verified for all graphs with at most 11 vertices. Further, the conjecture has been verified for regular graphs, trees, disconnected graphs, unit interval graphs, separable graphs with no pendant vertex, outer-planar graphs, and unicyclic graphs. We extend the list of graph classes for which the conjecture holds. We give a proof that bipartite permutation graphs are reconstructible.

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16

Jahanbani, Akbar, Hajar Shooshtari, and Yilun Shang. "Extremal trees for the Randić index." Acta Universitatis Sapientiae, Mathematica 14, no. 2 (December 1, 2022): 239–49. http://dx.doi.org/10.2478/ausm-2022-0016.

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Abstract Graph theory has applications in various fields due to offering important tools such as topological indices. Among the topological indices, the Randić index is simple and of great importance. The Randić index of a graph 𝒢 can be expressed as R ( G ) = ∑ x y ∈ Y ( G ) 1 τ ( x ) τ ( y ) R\left( G \right) = \sum\nolimits_{xy \in Y\left( G \right)} {{1 \over {\sqrt {\tau \left( x \right)\tau \left( y \right)} }}} , where 𝒴(𝒢) represents the edge set and τ(x) is the degree of vertex x. In this paper, considering the importance of the Randić index and applications two-trees graphs, we deter

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17

Panda, Swarup, and Dr Sukanta Pati. "On The Inverse Of A Class Of Bipartite Graphs With Unique Perfect Matchings." Electronic Journal of Linear Algebra 29 (September 20, 2015): 89–101. http://dx.doi.org/10.13001/1081-3810.2865.

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Let G be a simple, undirected graph and Gw be the weighted graph obtained from G by giving weights to its edges using a positive weight function w. A weighted graph Gw is said to be nonsingular if its adjacency matrix A(Gw) is nonsingular. In [9], Godsil has given a class $\mathcal{G }$of connected, unweighted, bipartite, nonsingular graphs G with a unique perfect matching, such that A(G)â1 is signature similar to a nonnegative matrix, that is, there exists a diagonal matrix D with diagonal entries Â±1 such that DA(G)â1D is nonnegative. The graph associated to the matrix DA(G)â1D is call

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18

Boaventura-Netto, Paulo Oswaldo. "Ranking graph edges by the weight of their spanning arborescences or trees." Pesquisa Operacional 28, no. 1 (April 2008): 59–73. http://dx.doi.org/10.1590/s0101-74382008000100004.

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A result based on a classic theorem of graph theory is generalized for edge-valued graphs, allowing determination of the total value of the spanning arborescences with a given root and containing a given arc in a directed valued graph. A corresponding result for undirected valued graphs is also presented. In both cases, the technique allows for a ranking of the graph edges by importance under this criterion. This ranking is proposed as a tool to determine the relative importance of the edges of a graph in network vulnerability studies. Some examples of application are presented.

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19

Li, Feng. "The Number of Spanning Trees in the Composition Graphs." Mathematical Problems in Engineering 2014 (2014): 1–5. http://dx.doi.org/10.1155/2014/613685.

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Using the composition of some existing smaller graphs to construct some large graphs, the number of spanning trees and the Laplacian eigenvalues of such large graphs are also closely related to those of the corresponding smaller ones. By using tools from linear algebra and matrix theory, we establish closed formulae for the number of spanning trees of the composition of two graphs with one of them being an arbitrary complete 3-partite graph and the other being an arbitrary graph. Our results extend some of the previous work, which depend on the structural parameters such as the number of verti

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20

Courcelle, Bruno. "Unfoldings and Coverings of Weighted Graphs." Fundamenta Informaticae 189, no. 1 (July 7, 2023): 1–47. http://dx.doi.org/10.3233/fi-222150.

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Coverings of undirected graphs are used in distributed computing, and unfoldings of directed graphs in semantics of programs. We study these two notions from a graph theoretical point of view so as to highlight their similarities, as they are both defined in terms of surjective graph homomorphisms. In particular, universal coverings and complete unfoldings are infinite trees that are regular if the initial graphs are finite. Regularity means that a tree has finitely many subtrees up to isomorphism. Two important theorems have been established by Leighton and Norris for coverings of finite grap

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21

Al-Janabi, H., and G. Bacsó. "Sanskruti Index of some Chemical Trees and Unicyclic Graphs." Journal of Physics: Conference Series 2287, no. 1 (June 1, 2022): 012005. http://dx.doi.org/10.1088/1742-6596/2287/1/012005.

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Abstract In chemical graph theory, topological indices are used to estimate the bio-activity of chemical compounds. The molecular graph models molecular compounds. A molecular graph represents the structural formula of a chemical compound relative to the graph. Its vertices represent the atoms of the compound, and its edges represent the chemical bonds. In 2017, Hosamani introduced the Sanskruti index S(G) and defined it as S(G) = Σuv∈E(G) ((δu ⋅ δv)/(δu + δv − 2))3, where δu is the sum of the degrees of all neighbors of the vertex u in G. The Sanskruti index displays a good connection with an

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22

Arockiaraj, Micheal, J. Nancy Delaila, and Jessie Abraham. "Optimal Wirelength of Balanced Complete Multipartite Graphs onto Cartesian Product of {Path, Cycle} and Trees." Fundamenta Informaticae 178, no. 3 (January 15, 2021): 187–202. http://dx.doi.org/10.3233/fi-2021-2003.

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In any interconnection network, task allocation plays a major role in the processor speed as fair distribution leads to enhanced performance. Complete multipartite networks serve well for this purpose as the task can be split into different partites which improves the degree of reliability of the network. Such an allocation process in the network can be done by means of graph embedding. The optimal wirelength of a graph embedding helps in the distribution of deterministic algorithms from the guest graph to other host graphs in order to incorporate its unique deterministic properties on that ch

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23

Manjula, V. "Graph Applications to Data Structures." Advanced Materials Research 433-440 (January 2012): 3297–301. http://dx.doi.org/10.4028/www.scientific.net/amr.433-440.3297.

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This paper presents a topic on Graph theory and its application to data Structures which I consider basic and useful to students in APPLIED MATHEMATICS and ENGINEERING.This paper gives an elementary introduction of Graph theory and its application to data structures. Elements of Graph theory are indispensable in almost all computer Science areas .It can be used in Some areas such as syntactic analysis, fault detection, diagnosis in computers and minimal path problems. The computer representation and manipulation of graph are also discussed so that certain algorithms can be included .A major th

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24

Zeen El Deen, Mohamed R., Walaa A. Aboamer, and Hamed M. El-Sherbiny. "The Complexity of the Super Subdivision of Cycle-Related Graphs Using Block Matrices." Computation 11, no. 8 (August 15, 2023): 162. http://dx.doi.org/10.3390/computation11080162.

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The complexity (number of spanning trees) in a finite graph Γ (network) is crucial. The quantity of spanning trees is a fundamental indicator for assessing the dependability of a network. The best and most dependable network is the one with the most spanning trees. In graph theory, one constantly strives to create novel structures from existing ones. The super subdivision operation produces more complicated networks, and the matrices of these networks can be divided into block matrices. Using methods from linear algebra and the characteristics of block matrices, we derive explicit formulas for

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25

Ramanujam, R., and Ramanathan S. Thinniyam. "Definability in first-order theories of graph orderings ⋆." Journal of Logic and Computation 30, no. 1 (January 2020): 403–20. http://dx.doi.org/10.1093/logcom/exaa017.

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Abstract We study definability in the first-order theory of graph order: i.e. the set of all isomorphism types of simple finite graphs ordered by either the minor, subgraph or induced subgraph relation. Natural graph families like cycles and trees are definable in these orders, as also notions like connectivity, maximum degree, etc. This machinery allows us to show mutual interpretability with arithmetic for all orders. We discuss implications for formalizing statements of graph theory in such theories of order. 1

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26

Wang, Shaohui, and Bing Wei. "Multiplicative Zagreb indices of cacti." Discrete Mathematics, Algorithms and Applications 08, no. 03 (August 2016): 1650040. http://dx.doi.org/10.1142/s1793830916500403.

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Let [Formula: see text] be multiplicative Zagreb index of a graph [Formula: see text]. A connected graph is a cactus graph if and only if any two of its cycles have at most one vertex in common, which is a generalization of trees and has been the interest of researchers in the field of material chemistry and graph theory. In this paper, we use a new tool to obtain the upper and lower bounds of [Formula: see text] for all cactus graphs and characterize the corresponding extremal graphs.

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27

Manimuthu, Yamuna, and Karthika Kumarasamy. "A Survey on Characterizing Trees Using Domination Number." Mathematics 10, no. 13 (June 22, 2022): 2173. http://dx.doi.org/10.3390/math10132173.

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Ever since the discovery of domination numbers by Claude Berge in the year 1958, graph domination has become an important domain in graph theory that has strengthened itself as a theory and has extended its contributions to various applications. Tree characterization is an important problem in graph domination. This survey focuses on presenting a collection of results on characterizing trees using domination number.

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28

Niwanthi, Wan. "Solving sweeping problem for trees in graph theory." Computer Science and Mathematical Modelling, no. 17/2023 (January 30, 2024): 29–33. http://dx.doi.org/10.5604/01.3001.0054.6288.

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We develop a theory to determine the search number of a graph that allows us to detect an intruder along an edge without limiting the visibility of adjacent vertices. The presented technique here will allow to express the sweep problem as a linear program using an existing formulation of a linear program designed for problems where capture occurs only at a vertex of a graph. We also provide a method to solve the sweep problem for any complex tree, utilizing a set of sub-trees of the tree.

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Hu, Yarong, and Qiongxiang Huang. "Quadratic Starlike Trees." Algebra Colloquium 30, no. 04 (November 28, 2023): 615–38. http://dx.doi.org/10.1142/s1005386723000470.

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We introduce the notion of a quadratic graph, which is a graph whose eigenvalues are integral or quadratic algebraic integral, and we determine nine infinite families of quadratic starlike trees, which are just all the quadratic starlike trees including integral starlike trees. Thus, the quadratic starlike trees are completely characterized. The expressions for characteristic polynomials of quadratic starlike trees are also given.

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PANDA, SWARUP. "Inverses of bicyclic graphs." Electronic Journal of Linear Algebra 32 (February 6, 2017): 217–31. http://dx.doi.org/10.13001/1081-3810.3322.

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A graph G is said to be nonsingular (resp., singular) if its adjacency matrix A(G) is nonsingular (resp., singular). The inverse of a nonsingular graph G is the unique weighted graph whose adjacency matrix is similar to the inverse of the adjacency matrix A(G) via a diagonal matrix of Â±1s. Consider connected bipartite graphs with unique perfect matchings such that the graph obtained by contracting all matching edges is also bipartite. In [C.D. Godsil. Inverses of trees. Combinatorica, 5(1):33â39, 1985.], Godsil proved that such graphs are invertible. He posed the question of characterizing

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Knor, Martin, Muhammad Imran, Muhammad Kamran Jamil та Riste Škrekovski. "Remarks on Distance Based Topological Indices for ℓ-Apex Trees". Symmetry 12, № 5 (12 травня 2020): 802. http://dx.doi.org/10.3390/sym12050802.

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A graph G is called an ℓ-apex tree if there exist a vertex subset A ⊂ V ( G ) with cardinality ℓ such that G − A is a tree and there is no other subset of smaller cardinality with this property. In the paper, we investigate extremal values of several monotonic distance-based topological indices for this class of graphs, namely generalized Wiener index, and consequently for the Wiener index and the Harary index, and also for some newer indices as connective eccentricity index, generalized degree distance, and others. For the one extreme value we obtain that the extremal graph is a join of a tre

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32

FRATI, FABRIZIO. "ON MINIMUM AREA PLANAR UPWARD DRAWINGS OF DIRECTED TREES AND OTHER FAMILIES OF DIRECTED ACYCLIC GRAPHS." International Journal of Computational Geometry & Applications 18, no. 03 (June 2008): 251–71. http://dx.doi.org/10.1142/s021819590800260x.

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It has been shown that there exist planar digraphs that require exponential area in every upward straight-line planar drawing. On the other hand, upward poly-line planar drawings of planar graphs can be realized in Θ(n2) area. In this paper we consider families of DAGs that naturally arise in practice, like DAGs whose underlying graph is a tree (directed trees), is a bipartite graph (directed bipartite graphs), or is an outerplanar graph (directed outerplanar graphs). Concerning directed trees, we show that optimal Θ(n log n) area upward straight-line/poly-line planar drawings can be construct

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33

Berry, Anne, and Geneviève Simonet. "Computing the Atom Graph of a Graph and the Union Join Graph of a Hypergraph." Algorithms 14, no. 12 (November 28, 2021): 347. http://dx.doi.org/10.3390/a14120347.

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The atom graph of a graph is a graph whose vertices are the atoms obtained by clique minimal separator decomposition of this graph, and whose edges are the edges of all possible atom trees of this graph. We provide two efficient algorithms for computing this atom graph, with a complexity in O(min(nωlogn,nm,n(n+m¯)) time, where n is the number of vertices of G, m is the number of its edges, m¯ is the number of edges of the complement of G, and ω, also denoted by α in the literature, is a real number, such that O(nω) is the best known time complexity for matrix multiplication, whose current valu

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Liu, Yuqing, and Nicholas A. Scoville. "The Realization Problem for Discrete Morse Functions on Trees." Algebra Colloquium 27, no. 03 (August 27, 2020): 455–68. http://dx.doi.org/10.1142/s1005386720000371.

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We introduce a new notion of equivalence of discrete Morse functions on graphs called persistence equivalence. Two functions are considered persistence equivalent if and only if they induce the same persistence diagram. We compare this notion of equivalence to other notions of equivalent discrete Morse functions. Then we compute an upper bound for the number of persistence equivalent discrete Morse functions on a fixed graph and show that this upper bound is sharp in the case where our graph is a tree. This is a version of the “realization problem” of the persistence map. We conclude with an e

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35

DARMANN, ANDREAS. "POPULAR SPANNING TREES." International Journal of Foundations of Computer Science 24, no. 05 (August 2013): 655–77. http://dx.doi.org/10.1142/s0129054113500226.

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Combinatorial Optimization is combined with Social Choice Theory when the goal is to decide on the quality of a spanning tree of an undirected graph. Given individual preferences over the edges of the graph, spanning trees are compared by means of a Condorcet criterion. The comparisons are based on scoring functions used in classic voting rules such as approval voting and Borda voting. In this work, we investigate the computational complexity involved in deciding on the quality of a spanning tree with respect to the different voting rules adapted. In particular, we draw the sharp separation li

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Lehtonen, Erkko, and Tamás Waldhauser. "Associative spectra of graph algebras I." Journal of Algebraic Combinatorics 53, no. 3 (May 2021): 613–38. http://dx.doi.org/10.1007/s10801-020-01010-w.

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AbstractAssociative spectra of graph algebras are examined with the help of homomorphisms of DFS trees. Undirected graphs are classified according to the associative spectra of their graph algebras; there are only three distinct possibilities: constant 1, powers of 2, and Catalan numbers. Associative and antiassociative digraphs are described, and associative spectra are determined for certain families of digraphs, such as paths, cycles, and graphs on two vertices.

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Tang, Liang Wen, Juan Liu, and Shuangwei Qin. "Construction of Equienergetic Trees." Match Communications in Mathematical and in Computer Chemistry 91, no. 2 (October 2023): 485–88. http://dx.doi.org/10.46793/match.91-2.485t.

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The energy E(G) of a graph G is the sum of the absolute values of all eigenvalues of G. Two graphs of the same order are said to be equienergetic if their energies are equal. As pointed out by Gutman, it is not known how to systematically construct any pair of equienergetic, non-cospectral trees until now. Inspired by the research of integral trees, we proposed a construction of infinite pairs of equienergetic trees of diameter 4.

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Kuhlmann, Marco, and Stephan Oepen. "Towards a Catalogue of Linguistic Graph Banks." Computational Linguistics 42, no. 4 (December 2016): 819–27. http://dx.doi.org/10.1162/coli_a_00268.

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Graphs exceeding the formal complexity of rooted trees are of growing relevance to much NLP research. Although formally well understood in graph theory, there is substantial variation in the types of linguistic graphs, as well as in the interpretation of various structural properties. To provide a common terminology and transparent statistics across different collections of graphs in NLP, we propose to establish a shared community resource with an open-source reference implementation for common statistics.

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Shang, Yilun. "On the number of spanning trees, the Laplacian eigenvalues, and the Laplacian Estrada index of subdivided-line graphs." Open Mathematics 14, no. 1 (January 1, 2016): 641–48. http://dx.doi.org/10.1515/math-2016-0055.

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Abstract As a generalization of the Sierpiński-like graphs, the subdivided-line graph Г(G) of a simple connected graph G is defined to be the line graph of the barycentric subdivision of G. In this paper we obtain a closed-form formula for the enumeration of spanning trees in Г(G), employing the theory of electrical networks. We present bounds for the largest and second smallest Laplacian eigenvalues of Г(G) in terms of the maximum degree, the number of edges, and the first Zagreb index of G. In addition, we establish upper and lower bounds for the Laplacian Estrada index of Г(G) based on the

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Bhattacharya, Anushree, and Madhumangal Pal. "A Fuzzy Graph Theory Approach to the Facility Location Problem: A Case Study in the Indian Banking System." Mathematics 11, no. 13 (July 4, 2023): 2992. http://dx.doi.org/10.3390/math11132992.

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A fuzzy graph G is stated to have a set of trees as its tree cover if all the vertices of G are in their union. The maximum weight tree in the tree cover is assumed to be the cost of a tree cover for a fuzzy graph. For an integer β>0, finding a set of trees to cover all the vertices of a graph with minimum cost and at most β number of spanning trees is known as the β-tree cover problem. Combining the tree-covering concept and facility location problem in a fuzzy environment for solving critical real-life problems in the recent era is a more fruitful approach. This issue strongly inspires us

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Du, Zhibin. "The sum of the first two largest signless Laplacian eigenvalues of trees and unicyclic graphs." Electronic Journal of Linear Algebra 35 (February 1, 2019): 449–67. http://dx.doi.org/10.13001/1081-3810.3405.

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Let $G$ be a graph on $n$ vertices with $e(G)$ edges. The sum of eigenvalues of graphs has been receiving a lot of attention these years. Let $S_2 (G)$ be the sum of the first two largest signless Laplacian eigenvalues of $G$, and define $f(G) = e (G) +3 - S_2 (G)$. Oliveira et al. (2015) conjectured that $f(G) \geqslant f(U_{n})$ with equality if and only if $G \cong U_n$, where $U_n$ is the $n$-vertex unicyclic graph obtained by attaching $n-3$ pendent vertices to a vertex of a triangle. In this paper, it is proved that $S_2(G) < e(G) + 3 -\frac{2}{n}$ when $G$ is a tree, or a unicyclic g

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MELLOR, BLAKE. "TREE DIAGRAMS FOR STRING LINKS." Journal of Knot Theory and Its Ramifications 15, no. 10 (December 2006): 1303–18. http://dx.doi.org/10.1142/s0218216506005159.

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In previous work, the author defined the intersection graph of a chord diagram associated with a string link (as in the theory of finite type invariants). In this paper, we classify the trees which can be obtained as intersection graphs of string link diagrams.

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Vasconcellos, Jucele França de Alencar, Edson Norberto Cáceres, Henrique Mongelli, Siang Wun Song, Frank Dehne, and Jayme Luiz Szwarcfiter. "New BSP/CGM algorithms for spanning trees." International Journal of High Performance Computing Applications 33, no. 3 (October 14, 2018): 444–61. http://dx.doi.org/10.1177/1094342018803672.

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Computing a spanning tree (ST) and a minimum ST (MST) of a graph are fundamental problems in graph theory and arise as a subproblem in many applications. In this article, we propose parallel algorithms to these problems. One of the steps of previous parallel MST algorithms relies on the heavy use of parallel list ranking which, though efficient in theory, is very time-consuming in practice. Using a different approach with a graph decomposition, we devised new parallel algorithms that do not make use of the list ranking procedure. We proved that our algorithms are correct, and for a graph [Form

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Dobrinen, Natasha. "The Ramsey theory of the universal homogeneous triangle-free graph." Journal of Mathematical Logic 20, no. 02 (January 28, 2020): 2050012. http://dx.doi.org/10.1142/s0219061320500129.

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The universal homogeneous triangle-free graph, constructed by Henson [A family of countable homogeneous graphs, Pacific J. Math. 38(1) (1971) 69–83] and denoted [Formula: see text], is the triangle-free analogue of the Rado graph. While the Ramsey theory of the Rado graph has been completely established, beginning with Erdős–Hajnal–Posá [Strong embeddings of graphs into coloured graphs, in Infinite and Finite Sets. Vol.[Formula: see text] , eds. A. Hajnal, R. Rado and V. Sós, Colloquia Mathematica Societatis János Bolyai, Vol. 10 (North-Holland, 1973), pp. 585–595] and culminating in work of S

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Mete, Filiz, Serife Buyukkose, Ozlem Cakir, and Ummugulsum Candeger. "Graphic representation of open and distance education history." Global Journal of Information Technology: Emerging Technologies 7, no. 3 (December 24, 2017): 92–98. http://dx.doi.org/10.18844/gjit.v7i3.2831.

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Nowadays, learning and instruction take place independent of time and space through the distance education system,wherein courses are conducted completely online through network technologies using interactive video -based instructional materials. This study examines the open and distance education system that was a part of the history of education in the Turkish republic first at universities, and then in high sch ools and secondary schools. It is aimed to narrate the history of open and distance education using graph theory trees in order to provide a better understanding of this process. Wit

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TRALDI, LORENZO. "Weighted Interlace Polynomials." Combinatorics, Probability and Computing 19, no. 1 (August 10, 2009): 133–57. http://dx.doi.org/10.1017/s0963548309990381.

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The interlace polynomials introduced by Arratia, Bollobás and Sorkin extend to invariants of graphs with vertex weights, and these weighted interlace polynomials have several novel properties. One novel property is a version of the fundamental three-term formulathat lacks the last term. It follows that interlace polynomial computations can be represented by binary trees rather than mixed binary–ternary trees. Binary computation trees provide a description ofq(G) that is analogous to the activities description of the Tutte polynomial. IfGis a tree or forest then these ‘algorithmic activities’ a

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Das, Kinkar, and SHAOWEI SUN. "Extremal graph on normalized Laplacian spectral radius and energy." Electronic Journal of Linear Algebra 29 (September 20, 2015): 237–53. http://dx.doi.org/10.13001/1081-3810.3263.

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Let $G=(V,\,E)$ be a simple graph of order $n$ and the normalized Laplacian eigenvalues $\rho_1\geq \rho_2\geq \cdots\geq\rho_{n-1}\geq \rho_n=0$. The normalized Laplacian energy (or Randi\'c energy) of $G$ without any isolated vertex is defined as $$RE(G)=\sum_{i=1}^{n}|\rho_i-1|.$$ In this paper, a lower bound on $\rho_1$ of connected graph $G$ ($G$ is not isomorphic to complete graph) is given and the extremal graphs (that is, the second minimal normalized Laplacian spectral radius of connected graphs) are characterized. Moreover, Nordhaus-Gaddum type results for $\rho_1$ are obtained. Rece

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Göbel, Andreas, J. A. Gregor Lagodzinski, and Karen Seidel. "Counting Homomorphisms to Trees Modulo a Prime." ACM Transactions on Computation Theory 13, no. 3 (September 30, 2021): 1–33. http://dx.doi.org/10.1145/3460958.

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Many important graph-theoretic notions can be encoded as counting graph homomorphism problems, such as partition functions in statistical physics, in particular independent sets and colourings. In this article, we study the complexity of # p H OMS T O H , the problem of counting graph homomorphisms from an input graph to a graph H modulo a prime number p . Dyer and Greenhill proved a dichotomy stating that the tractability of non-modular counting graph homomorphisms depends on the structure of the target graph. Many intractable cases in non-modular counting become tractable in modular counting

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Domagalski, Rachel, and Sivaram Narayan. "Tree Cover Number and Maximum Semidefinite Nullity of Some Graph Classes." Electronic Journal of Linear Algebra 36, no. 36 (September 30, 2020): 678–93. http://dx.doi.org/10.13001/ela.2020.5319.

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Let $G$ be a graph with a vertex set $V$ and an edge set $E$ consisting of unordered pairs of vertices. The tree cover number of $G$, denoted $\tau(G)$, is the minimum number of vertex disjoint simple trees occurring as induced subgraphs of $G$ that cover all the vertices of $G$. In this paper, the tree cover number of a line graph $\tau(L(G))$ is shown to be equal to the path number $\pi(G)$ of $G$. Also, the tree cover numbers of shadow graphs, corona and Cartesian product of two graphs are found. The graph parameter $\tau(G)$ is related to another graph parameter $M_+(G)$, called the

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Nandini, G. Kirithiga, R. Sundara Rajan, A. Arul Shantrinal, T. M. Rajalaxmi, Indra Rajasingh, and Krishnan Balasubramanian. "Topological and Thermodynamic Entropy Measures for COVID-19 Pandemic through Graph Theory." Symmetry 12, no. 12 (December 2, 2020): 1992. http://dx.doi.org/10.3390/sym12121992.

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Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) has caused the global pandemic, coronavirus disease-2019 (COVID-19) which has resulted in 60.4 million infections and 1.42 million deaths worldwide. Mathematical models as an integral part of artificial intelligence are designed for contact tracing, genetic network analysis for uncovering the biological evolution of the virus, understanding the underlying mechanisms of the observed disease dynamics, evaluating mitigation strategies, and predicting the COVID-19 pandemic dynamics. This paper describes mathematical techniques to exploit

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