Tree Graph Research Articles

PT-symmetric solitons in networks: a metric graph based approach

Abstract We propose a model for networks approving PT-symmetric standing solitons. The model is described in terms of the nonlinear Schrödinger equation on metric graphs, with complex PT-symmetric potential given on each bond. Exact soliton solutions of the problem are obtained. The stability analysis of the PT-symmetric standing solitons on the star graph is studied. Extension for other graph topologies is demonstrated by considering a metric tree graph.

SPANNING TREES IN $\mathbb {Z}$ -COVERS OF A FINITE GRAPH AND MAHLER MEASURES

Abstract Using the special value at $u=1$ of Artin–Ihara L-functions, we associate to every $\mathbb {Z}$ -cover of a finite connected graph a polynomial, which we call the Ihara polynomial. We show that the number of spanning trees for the finite intermediate graphs of such a cover can be expressed in terms of the Pierce–Lehmer sequence associated to a factor of the Ihara polynomial. This allows us to express the asymptotic growth of the number of spanning trees in terms of the Mahler measure of this polynomial. Specialising to the situation where the base graph is a bouquet or the dumbbell graph gives us back previous results in the literature for circulant and I-graphs (including the generalised Petersen graphs). We also express the p-adic valuation of the number of spanning trees of the finite intermediate graphs in terms of the p-adic Mahler measure of the Ihara polynomial. When applied to a particular $\mathbb {Z}$ -cover, our result gives us back Lengyel’s calculation of the p-adic valuations of Fibonacci numbers.

Counting spanning trees of multiple complete split-like graph containing a given spanning forest

The multiple complete split-like graph MCSb,sa is the join of an empty graph K‾a and s copies of complete graph Kb. In this article, we obtain the formulas for the number of spanning trees of MCSb,sa containing a given spanning forest when s=1 and 2. Particularly, when s=1, our result derives the number of spanning trees of complete split graph containing a given spanning forest, thereby extending Moon's result [19].

BINARY FRUIT FLY OPTIMIZATION ALGORITHM FOR COUNTING THE NUMBER OF SPANNING TREES IN NETWORKS

The NP-hard problem of determining the number of spanning trees of graphs is examined in this paper. A spanning tree of a graph is a tree such that (i) it is a subgraph of (i.e., that includes only edges from ), and (ii) it includes every vertex of . The most classical interest concerning a spanning tree is the number of spanning trees or the complexity of the graph denoted by We propose the first attempt to use a binary version of the fruit fly optimization algorithm (BFFOA) to compute the minimal spanning tree of a graph in a heuristic manner. The fruit fly of BFFOA is binary encoded and used to represent which one of the vertices of the graph belongs to the spanning tree of . The feasibility is enforced by repairing the fruit fly such that an extra vertex created from vertices of is added to , and this repairing process is repeated until becomes the spanning tree of . Theoretically computed graph results are used to compare the proposed BFFOA against competing techniques. The proposed BFFOA performs better than the binary Grey Wolf Optimizer (BGWO), the binary Particle Swarm Optimizer (BPSO), the binary Whale Optimizer (BWO), and the binary Multi-verse Optimization (BMVO) algorithms, according to analysis and computational results.

The number of spanning trees in K n -chain and ring graphs

The problem of counting spanning trees of graphs or networks is a fundamental and crucial area of research in combinatorics, while has numerous important applications in statistical physics, network theory and theoretical computer science. Very recently, Kosar, Zaman, Ali and Ullah obtained a nice formula on the number of spanning trees of a K 5-chain network K5ℓ constructed by connecting ℓcopys of complete graphs K 5. They made extensive use of matrix theory and spectral graph theory, especially the normalized Laplacian of graphs. In this paper, by using a rather simple and more physical treatment (the mesh-star transformation in electrical network) without any linear algebra, we generalize their result to K n -chain graphs and K n -ring graphs. The results show that there is a simple relation between the number of spanning trees of the K n -chain graph Lnℓ and the K n -ring graph Cnℓ . We also calculate the corresponding tree entropy (or so called ‘the asymptotic growth constant’) and find that the tree entropy of the corresponding K n -chain graphs and K n -ring graphs are totally the same.

Bottleneck matrices of maximal outerplanar graphs with isomorphic underlying trees

In this paper, we consider the entries of the bottleneck matrices of maximal outerplanar graphs with isomorphic underlying trees. We show how the entries of the bottleneck matrix are perturbed when we modify a maximal outerplanar graph into a nonisomorphic maximal outerplanar graph whose underlying tree is isomorphic to the underlying tree of the original maximal outerplanar graph. We illustrate our results with examples.

Extremal values on the Kirchhoff index of the line graph of trees

The computation of resistance distance and the Kirchhoff index is a classical problem that has been extensively investigated by numerous mathematicians, physicists, and scientists. Consider a simple connected graph G with vertex set V(G) and edge set E(G). If we replace each edge of G with a resistor of 1 ohm resistance, we create an electrical network N; in that case, the distance between any two nodes between network N is called resistance distance. The Kirchhoff index is a mathematical term that quantifies the complexity of a graph; it is defined as the total resistance distance among each pair of vertices in G. The line graph LG is constructed from G by swapping out the edges for vertices; in LG, if two vertices share endpoints in G, then they are connected in LG. In this study, extremal values for the Kirchhoff index of the line graph of trees are calculated. Further, we will also calculate the Kirchhoff index for the line graph of some special trees and establish the relationship between them.

The Sombor Indices of Banana Tree Graph and Fractal Tree Type Dendrimer

Topological descriptors are graph-theoretical metrics that allow one to describe a molecular structure’s underlying connectedness. Degree-based topological descriptors have been investigated widely and are associated with several chemical characteristics. The examination of graph entropy indices as a gauge for the intricacy of the underlying connection and as a means of characterizing structural attributes has also gained significance.The present study utilizes the graph-theory-based edge partition method to examine Sombor variants for the Banana Tree Graph, Fractal, and Cayley Tree Type Dendrimers. We have employed Shannon’s entropy model to determine the graph-based entropy of these graphs and produced Sombor indices.

Multilevel interpolations for the generalized Nikishin system on a tree graph

We study a multilevel interpolation problem for a system of Markov functions defined by a tree graph. The normality of all indices has been proven. Asymptotic properties are studied in terms of the vector equilibrium problem for logarithmic potential.

Paired coalition in graphs

A paired coalition in a graph G = ( V , E ) consists of two disjoint sets of vertices V 1 and V 2 , neither of which is a paired dominating set but whose union V 1 ∪ V 2 is a paired dominating set. A paired coalition partition (abbreviated pc-partition) in a graph G is a vertex partition π = { V 1 , V 2 , … , V k } such that each set V i of π is not a paired dominating set but forms a paired coalition with another set V j ∈ π . The paired coalition graph PCG ( G , π ) of the graph G with the pc-partition π of G, is the graph whose vertices correspond to the sets of π , and two vertices V i and V j are adjacent in PCG ( G , π ) if and only if their corresponding sets V i and V j form a paired coalition in G. In this paper, we initiate the study of paired coalition partitions and paired coalition graphs. In particular, we determine the paired coalition number of paths and cycles, obtain some results on paired coalition partitions in trees and characterize pair coalition graphs of paths, cycles and trees. We also characterize triangle-free graphs G of order n with PC ( G ) = n and unicyclic graphs G of order n with PC ( G ) = n − 2 .

DC-RST: a parallel algorithm for random spanning trees in network analytics

The Mantel Test, discovered in the 1960s, determines whether two distance metrics on a network are related. More recently, DimeCost, an equivalent test with improved computational complexity, was proposed. It was based on computing a random spanning tree of a complete graph on n vertices—the spanning tree was computed using Wilson’s random walk algorithm. In this paper, we describe DC-RST, a parallel, divide-and-conquer random walk algorithm to further speed up this computation. Relative to Wilson’s sequential random-walk algorithm, on a system with 48 cores, DC-RST was up to 4X faster when first creating random partitions and up to 20X faster without this sub-step. DC-RST is shown to be a suitable replacement for the Mantel and DimeCost tests through a combination of theoretical and statistical results.

Structure and position-aware graph neural network for airway labeling

We present a novel graph-based approach for labeling the anatomical branches of a given airway tree segmentation. The proposed method formulates airway labeling as a branch classification problem in the airway tree graph, where branch features are extracted using convolutional neural networks and enriched using graph neural networks. Our graph neural network is structure-aware by having each node aggregate information from its local neighbors and position-aware by encoding node positions in the graph.We evaluated the proposed method on 220 airway trees from subjects with various severity stages of Chronic Obstructive Pulmonary Disease (COPD). The results demonstrate that our approach is computationally efficient and significantly improves branch classification performance than the baseline method. The overall average accuracy of our method reaches 91.18% for labeling 18 segmental airway branches, compared to 83.83% obtained by the standard CNN method and 87.37% obtained by the existing method. Furthermore, the reader study done on an additional set of 40 subjects shows that our algorithm performs comparably to human experts in labeling segmental-airways. We published our source code at https://github.com/DIAGNijmegen/spgnn. The proposed algorithm is also publicly available at https://grand-challenge.org/algorithms/airway-anatomical-labeling/.

Correction to “Sharp threshold for embedding balanced spanning trees in random geometric graphs”

Correction to “Sharp threshold for embedding balanced spanning trees in random geometric graphs”

The Complexity of Octopus Graph, Friendship Graph, and Snail Graph

Graphs are basic structures that represent objects with nodes and relationships between objects with edges. Trees are one of the parts studied in graph theory along with finding the number of spanning trees of a graph such as octopus graph, friendship graph, and snail graph. The complexity of an octopus graph is strongly dependent on the number and length of tentacles, the complexity of a friendship graph is dependent on the number of triangle cycles, and the complexity of a snail graph is dependent on the number of edges and vertices located in the shell-like part of the snail. To calculate the number of spanning trees (τ(G)) of a graph, various calculations can be used, such as the extension of Kirchhoff's formula. The extension of Kirchhoff's formula uses the determinant of the adjacency matrix and degree matrix of the graph complement of a graph. Therefore, this research applies the extension of Kirchhoff's formula to obtain the complexity of octopus graph, friendship graph, and snail graph. From the analysis, it is obtained that for any n≥2, the number of spanning trees of octopus graph and friendship graph are τ(On )=1/5 √5 [((3+√5)/2)^n-((3-√5)/2)^n ] and τ(Fn )=3^n and the number of spanning trees of snail graph is τ(SIn )=2^(n+2)+3n∙2^(n-1) for n≥1.

Approximation algorithms for maximum weighted internal spanning trees in regular graphs and subdivisions of graphs

Abstract Let $G$ be a vertex-weighted connected graph of $n$ vertices and let $T$ be a spanning tree of $G$. We call $T$ a maximum weighted internal spanning tree of $G$ if the sum of the weights of the internal vertices of $T$ is the maximum over all spanning trees of $G$. The maximum weighted internal spanning tree (MaxwIST) problem asks to find such a spanning tree $T$ of $G$. The problem is NP-hard. We give an $O(dn)$ time approximation algorithm for $d$-regular graphs of $n=|V|$ vertices that computes a spanning tree with total weight of the internal vertices is at least $\frac{\beta _{d}}{\beta _{d} +d-2} - \epsilon $ of the total weight of all the vertices of the graph for any $\epsilon>0$, where $\beta _{d} = (d-1)H_{d-1}$, and $H_{d-1} = \sum _{i=1}^{d-1} i^{-1}$ is the $(d-1)$th harmonic number. For every $d \geq 3$ and $n_{0} \geq 1$, we show the construction of a $d$-regular graph of at least $n_{0}$ vertices, such that for any of its spanning trees, $\frac{w(I)}{w(V)}\le \frac{d}{d+1}$ holds. We give an $O(dn)$ time approximation algorithm for subdivisions of $d$-regular graphs, where the ratio of the internal weight of the spanning tree with the total vertex weight of the graph is at least $\frac{d-1}{2d-3} - \epsilon $ for $\epsilon>0$. We extend our study to $x$-subdivisions of Hamiltonian and hypoHamiltonian graphs, where each edge of the original Hamiltonian or hypoHamiltonian graph has been subdivided at least $x$ times. For those two graph classes, we show that there exists a spanning tree with internal vertex weight at least $1-\frac{2}{x-1}$ of the total vertex weight of the graph. Furthermore, we give $O(n)$ time algorithm for $x$-subdivisions of biconnected outerplanar graphs and $4$-connected planar graphs to achieve the above bound.

The 4-Set Tree Connectivity of Folded Hypercube

The [Formula: see text]-set tree connectivity, as a natural extension of classical connectivity, is a very important index to evaluate the fault-tolerance of interconnection networks. Let [Formula: see text] be a connected graph and a subset [Formula: see text], an [Formula: see text]-tree of graph [Formula: see text] is a tree [Formula: see text] that contains all the vertices of [Formula: see text] and [Formula: see text]. Two [Formula: see text]-trees [Formula: see text] and [Formula: see text] are internally disjoint if and only if [Formula: see text] and [Formula: see text]. The cardinality of maximum internally disjoint [Formula: see text]-trees is defined as [Formula: see text], and the [Formula: see text]-set tree connectivity is defined by [Formula: see text]. In this paper, we show that the [Formula: see text]-set tree connectivity of folded hypercube when [Formula: see text], that is, [Formula: see text], where [Formula: see text] is folded hypercube for [Formula: see text].

Loops of loops expansion in the amplituhedron

We study a novel geometric expansion for scattering amplitudes in the planar sector of N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 super Yang-Mills theory, in the context of the Amplituhedron which reproduces the all-loop integrand as a canonical differential form on the positive geometry. In a paper by Arkani-Hamed, Henn and one of the authors, it was shown that this result can be recast in terms of negative geometries with a certain hierarchy of loops (closed cycles) in the space of loop momenta, represented by lines in momentum twistor space. One can then calculate an all-loop order result in the approximation where only tree graphs in the space of all loops are considered. Furthermore, using differential equation methods, it is possible to calculate and resum integrated expressions and obtain strong coupling results. In this paper, we provide a more general framework for the ‘loops of loops’ expansion and outline a powerful method for the determination of differential forms for higher-order geometries. We solve the problem completely for graphs with one internal cycle, but the method can be used more generally for other geometries.

On the Eccentric Graph of Trees

We consider the eccentric graph of a graph G , denoted by e c c ( G ) , which has the same vertex set as G , and two vertices in the eccentric graph are adjacent if and only if their distance in G is equal to the eccentricity of one of them. In this paper, we present a fundamental requirement for the isomorphism between e c c ( G ) and the complement of G , and show that the previous necessary condition given in the literature is inadequate. Also, we obtain that the diameter of e c c ( T ) is at most 3 for any tree and get some characterizations of the eccentric graph of trees.

Polyalphabetic cipher cryptosystem application in making anti-hacker passwords learning management systems in junior high schools

Research on developing an anti-hacker network security system for the learning management system was conducted at SMP Plus Darus Salam. Learning activities at SMP Plus Darus Salam use LMS as a media, which is a program to maximize learning activities at school. its also has risks, it is related to the security of students data work, so network security is needed. Network security used within computer networks to facilitate secure communication in different network entities based on predefined security protocols. On the other hand, the Learning Management System (LMS) is a software platform designed for online educational activities, e-learning programs, and educational resources, simplifying student engagement in learning processes. This study employs applied research methodology, focusing on a single tribune graph, and involves a labeling process for vertices and edges to construct a graph tree, resulting in the generation of ciphertext. This research aims to develop anti-hacker passwords in the SMP Plus Darusalam LMS using a polyalphabetic cipher cryptosystem. The research utilizes a data set comprising 24 LMS account samples. Data collection techniques encompass observation, interviews, distribution of questionnaires, and documentation, while data analysis involves tabulation and the generation of ciphertext codes in the form of usernames and passwords. Based on the results, it shows a total score of 2,096 with an average of 91.13 or a security percentage of 91.13%, which shows that it is successful and useful in the SMP Plus Darus Salam learning management system (LMS) account. Thus, it can be concluded that the secjar (network security) network security efforts made were successful and provided significant benefits in securing learning management system (LMS) accounts at SMP Plus Darus Salam. Therefore, the researcher suggests to the readers to conduct research on the development of anti-hacker network security using Tribun graphs with ???? = 1 and ???? = 2.

Document-level biomedical relation extraction via hierarchical tree graph and relation segmentation module.

Biomedical relation extraction at the document level (Bio-DocRE) involves extracting relation instances from biomedical texts that span multiple sentences, often containing various entity concepts such as genes, diseases, chemicals, variants, etc. Currently, this task is usually implemented based on graphs or transformers. However, most work directly models entity features to relation prediction, ignoring the effectiveness of entity pair information as an intermediate state for relation prediction. In this article, we decouple this task into a three-stage process to capture sufficient information for improving relation prediction. We propose an innovative framework HTGRS for Bio-DocRE, which constructs a hierarchical tree graph (HTG) to integrate key information sources in the document, achieving relation reasoning based on entity. In addition, inspired by the idea of semantic segmentation, we conceptualize the task as a table-filling problem and develop a relation segmentation (RS) module to enhance relation reasoning based on the entity pair. Extensive experiments on three datasets show that the proposed framework outperforms the state-of-the-art methods and achieves superior performance. Our source code is available at https://github.com/passengeryjy/HTGRS.